<div class="xblock xblock-public_view xblock-public_view-vertical" data-usage-id="block-v1:MITx+15.053x+3T2016+type@vertical+block@f736c4f0a7674b82bcd7e6750e3bf82a" data-block-type="vertical" data-graded="False" data-course-id="course-v1:MITx+15.053x+3T2016" data-has-score="False" data-request-token="6a146286852711ef8f57023e4242adeb" data-init="VerticalStudentView" data-runtime-class="LmsRuntime" data-runtime-version="1"> <h2 class="hd hd-2 unit-title">Problem 1</h2> <div class="vert-mod"> <div class="vert vert-0" data-id="block-v1:MITx+15.053x+3T2016+type@problem+block@730cedbc07834f1490c3ed1fbe4d2b88"> <div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-usage-id="block-v1:MITx+15.053x+3T2016+type@problem+block@730cedbc07834f1490c3ed1fbe4d2b88" data-block-type="problem" data-graded="False" data-course-id="course-v1:MITx+15.053x+3T2016" data-has-score="True" data-request-token="6a146286852711ef8f57023e4242adeb" data-init="XBlockToXModuleShim" data-runtime-class="LmsRuntime" data-runtime-version="1"> <script type="json/xblock-args" class="xblock-json-init-args"> {"xmodule-type": "Problem"} </script> <div id="problem_730cedbc07834f1490c3ed1fbe4d2b88" class="problems-wrapper" role="group" aria-labelledby="730cedbc07834f1490c3ed1fbe4d2b88-problem-title" data-problem-id="block-v1:MITx+15.053x+3T2016+type@problem+block@730cedbc07834f1490c3ed1fbe4d2b88" data-url="/courses/course-v1:MITx+15.053x+3T2016/xblock/block-v1:MITx+15.053x+3T2016+type@problem+block@730cedbc07834f1490c3ed1fbe4d2b88/handler/xmodule_handler" data-problem-score="0" data-problem-total-possible="0" data-attempts-used="0" data-content=" &lt;h3 class=&#34;hd hd-3 problem-header&#34; id=&#34;730cedbc07834f1490c3ed1fbe4d2b88-problem-title&#34; aria-describedby=&#34;block-v1:MITx+15.053x+3T2016+type@problem+block@730cedbc07834f1490c3ed1fbe4d2b88-problem-progress&#34; tabindex=&#34;-1&#34;&gt; PART A &lt;/h3&gt; &lt;div class=&#34;problem-progress&#34; id=&#34;block-v1:MITx+15.053x+3T2016+type@problem+block@730cedbc07834f1490c3ed1fbe4d2b88-problem-progress&#34;&gt;&lt;/div&gt; &lt;div class=&#34;problem&#34;&gt; &lt;div&gt; &lt;p&gt;Consider the following nonlinear constrained minimization problem:&lt;/p&gt; &lt;span&gt; &lt;p class=&#34;equation&#34;&gt;\[\left.\begin{array}{rrcl} \min &amp;amp; \sqrt{x_2} \\ \text {s.t.:} &amp;amp; &amp;amp; &amp;amp; \\ &amp;amp; x_2 \geq (\frac{5}{2} x_1)^3 \\ &amp;amp; x_2 \geq (-\frac{3}{2}x_1 + 1)^3 \\ &amp;amp; x_2 \geq 0 \end{array}\right\}\]&lt;/p&gt; &lt;/span&gt; &lt;p&gt;A plot can be found below&lt;/p&gt; &lt;img width=&#34;500&#34; src=&#34;/assets/courseware/v1/aad3a0beffcba220c6528ef299dd36b2/asset-v1:MITx+15.053x+3T2016+type@asset+block/R6P1plot.png&#34;/&gt; &lt;p&gt; &lt;i&gt;Hint: Several algorithms will work, but we suggest the method of moving asymptotes, the details of which are not necessary to understand for the time being.&lt;/i&gt; &lt;/p&gt; &lt;p&gt;To two decimals, what is the objective value?&lt;/p&gt; &lt;p&gt;You can solve using spreadsheet optimization or using Julia. If you use Julia, you will need the following additional syntax &lt;pre&gt;&lt;code&gt; Pkg.add(&#34;NLopt&#34;) yourModelVariable=Model(solver=NLoptSolver(algorithm=:LD_MMA)) @NLobjective(yourModelVariable, Max/Min, Function) @NLconstraint(yourModelVariable, Inequality)) setvalue(variable, initialValue) &lt;/code&gt;&lt;/pre&gt;&lt;/p&gt; &lt;div class=&#34;wrapper-problem-response&#34; tabindex=&#34;-1&#34; aria-label=&#34;Question 1&#34; role=&#34;group&#34;&gt;&lt;div id=&#34;formulaequationinput_730cedbc07834f1490c3ed1fbe4d2b88_2_1&#34; class=&#34;inputtype formulaequationinput&#34;&gt; &lt;div class=&#34;unanswered&#34;&gt; &lt;input type=&#34;text&#34; name=&#34;input_730cedbc07834f1490c3ed1fbe4d2b88_2_1&#34; id=&#34;input_730cedbc07834f1490c3ed1fbe4d2b88_2_1&#34; data-input-id=&#34;730cedbc07834f1490c3ed1fbe4d2b88_2_1&#34; value=&#34;&#34; aria-describedby=&#34;status_730cedbc07834f1490c3ed1fbe4d2b88_2_1&#34; size=&#34;20&#34;/&gt; &lt;span class=&#34;trailing_text&#34; id=&#34;trailing_text_730cedbc07834f1490c3ed1fbe4d2b88_2_1&#34;/&gt; &lt;span class=&#34;status unanswered&#34; id=&#34;status_730cedbc07834f1490c3ed1fbe4d2b88_2_1&#34; data-tooltip=&#34;Not yet answered.&#34;&gt; &lt;span class=&#34;sr&#34;&gt;unanswered&lt;/span&gt;&lt;span class=&#34;status-icon&#34; aria-hidden=&#34;true&#34;/&gt; &lt;/span&gt; &lt;p id=&#34;answer_730cedbc07834f1490c3ed1fbe4d2b88_2_1&#34; class=&#34;answer&#34;/&gt; &lt;div id=&#34;input_730cedbc07834f1490c3ed1fbe4d2b88_2_1_preview&#34; class=&#34;equation&#34;&gt; \(\) &lt;img src=&#34;/static/images/spinner.bc34f953403f.gif&#34; class=&#34;loading&#34; alt=&#34;Loading&#34;/&gt; &lt;/div&gt; &lt;/div&gt; &lt;div class=&#34;script_placeholder&#34; data-src=&#34;/static/js/capa/src/formula_equation_preview.b1967ab28c31.js&#34;/&gt; &lt;/div&gt;&lt;/div&gt; &lt;div class=&#34;solution-span&#34;&gt; &lt;span id=&#34;solution_730cedbc07834f1490c3ed1fbe4d2b88_solution_1&#34;/&gt; &lt;/div&gt;&lt;/div&gt; &lt;div class=&#34;action&#34;&gt; &lt;input type=&#34;hidden&#34; name=&#34;problem_id&#34; value=&#34;PART A&#34; /&gt; &lt;div class=&#34;submit-attempt-container&#34;&gt; &lt;button type=&#34;button&#34; class=&#34;submit btn-brand&#34; data-submitting=&#34;Submitting&#34; data-value=&#34;Submit&#34; data-should-enable-submit-button=&#34;True&#34; aria-describedby=&#34;submission_feedback_730cedbc07834f1490c3ed1fbe4d2b88&#34; &gt; &lt;span class=&#34;submit-label&#34;&gt;Submit&lt;/span&gt; &lt;/button&gt; &lt;div class=&#34;submission-feedback&#34; id=&#34;submission_feedback_730cedbc07834f1490c3ed1fbe4d2b88&#34;&gt; &lt;span class=&#34;sr&#34;&gt;Some problems have options such as save, reset, hints, or show answer. These options follow the Submit button.&lt;/span&gt; &lt;/div&gt; &lt;/div&gt; &lt;div class=&#34;problem-action-buttons-wrapper&#34;&gt; &lt;/div&gt; &lt;/div&gt; &lt;div class=&#34;notification warning notification-gentle-alert is-hidden&#34; tabindex=&#34;-1&#34;&gt; &lt;span class=&#34;icon fa fa-exclamation-circle&#34; aria-hidden=&#34;true&#34;&gt;&lt;/span&gt; &lt;span class=&#34;notification-message&#34; aria-describedby=&#34;730cedbc07834f1490c3ed1fbe4d2b88-problem-title&#34;&gt; &lt;/span&gt; &lt;div class=&#34;notification-btn-wrapper&#34;&gt; &lt;button type=&#34;button&#34; class=&#34;btn btn-default btn-small notification-btn review-btn sr&#34;&gt;Review&lt;/button&gt; &lt;/div&gt; &lt;/div&gt; &lt;div class=&#34;notification warning notification-save is-hidden&#34; tabindex=&#34;-1&#34;&gt; &lt;span class=&#34;icon fa fa-save&#34; aria-hidden=&#34;true&#34;&gt;&lt;/span&gt; &lt;span class=&#34;notification-message&#34; aria-describedby=&#34;730cedbc07834f1490c3ed1fbe4d2b88-problem-title&#34;&gt;None &lt;/span&gt; &lt;div class=&#34;notification-btn-wrapper&#34;&gt; &lt;button type=&#34;button&#34; class=&#34;btn btn-default btn-small notification-btn review-btn sr&#34;&gt;Review&lt;/button&gt; &lt;/div&gt; &lt;/div&gt; &lt;div class=&#34;notification general notification-show-answer is-hidden&#34; tabindex=&#34;-1&#34;&gt; &lt;span class=&#34;icon fa fa-info-circle&#34; aria-hidden=&#34;true&#34;&gt;&lt;/span&gt; &lt;span class=&#34;notification-message&#34; aria-describedby=&#34;730cedbc07834f1490c3ed1fbe4d2b88-problem-title&#34;&gt;Answers are displayed within the problem &lt;/span&gt; &lt;div class=&#34;notification-btn-wrapper&#34;&gt; &lt;button type=&#34;button&#34; class=&#34;btn btn-default btn-small notification-btn review-btn sr&#34;&gt;Review&lt;/button&gt; &lt;/div&gt; &lt;/div&gt; &lt;/div&gt; " data-graded="False"> <p class="loading-spinner"> <i class="fa fa-spinner fa-pulse fa-2x fa-fw"></i> <span class="sr">Loading&hellip;</span> </p> </div> </div> </div> </div> </div>

<div class="xblock xblock-public_view xblock-public_view-vertical" data-usage-id="block-v1:MITx+15.053x+3T2016+type@vertical+block@934137a1e1004e368972ca7dcf485f59" data-block-type="vertical" data-graded="False" data-course-id="course-v1:MITx+15.053x+3T2016" data-has-score="False" data-request-token="6a146286852711ef8f57023e4242adeb" data-init="VerticalStudentView" data-runtime-class="LmsRuntime" data-runtime-version="1"> <h2 class="hd hd-2 unit-title">Problem 2</h2> <div class="vert-mod"> <div class="vert vert-0" data-id="block-v1:MITx+15.053x+3T2016+type@problem+block@8706ba6e7738497194ac73239f763fee"> <div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-usage-id="block-v1:MITx+15.053x+3T2016+type@problem+block@8706ba6e7738497194ac73239f763fee" data-block-type="problem" data-graded="False" data-course-id="course-v1:MITx+15.053x+3T2016" data-has-score="True" data-request-token="6a146286852711ef8f57023e4242adeb" data-init="XBlockToXModuleShim" data-runtime-class="LmsRuntime" data-runtime-version="1"> <script type="json/xblock-args" class="xblock-json-init-args"> {"xmodule-type": "Problem"} </script> <div id="problem_8706ba6e7738497194ac73239f763fee" class="problems-wrapper" role="group" aria-labelledby="8706ba6e7738497194ac73239f763fee-problem-title" data-problem-id="block-v1:MITx+15.053x+3T2016+type@problem+block@8706ba6e7738497194ac73239f763fee" data-url="/courses/course-v1:MITx+15.053x+3T2016/xblock/block-v1:MITx+15.053x+3T2016+type@problem+block@8706ba6e7738497194ac73239f763fee/handler/xmodule_handler" data-problem-score="0" data-problem-total-possible="0" data-attempts-used="0" data-content=" &lt;h3 class=&#34;hd hd-3 problem-header&#34; id=&#34;8706ba6e7738497194ac73239f763fee-problem-title&#34; aria-describedby=&#34;block-v1:MITx+15.053x+3T2016+type@problem+block@8706ba6e7738497194ac73239f763fee-problem-progress&#34; tabindex=&#34;-1&#34;&gt; PART A &lt;/h3&gt; &lt;div class=&#34;problem-progress&#34; id=&#34;block-v1:MITx+15.053x+3T2016+type@problem+block@8706ba6e7738497194ac73239f763fee-problem-progress&#34;&gt;&lt;/div&gt; &lt;div class=&#34;problem&#34;&gt; &lt;div&gt; &lt;p&gt;An investor has $5000 and two potential investments. Let \(x_j \text{ for } j = 1 \text{ and } j = 2 \) denote his allocation to investment \(j \) in thousands of dollars. From historical data, investments 1 and 2 have an expected annual return of 20 and 16 percent, respectively. Also, the total risk involved with investments 1 and 2, as measured by the variance of total return, is given by \(2x_1^2 +x_2^2 +(x_1+x_2)^2\), so that risk increases with total investment and with the amount of each individual investment. The investor would like to maximize his expected return and at the same time minimize his risk. Clearly, both of these objectives cannot, in general, be satisfied simultaneously. There are several possible approaches. For example, he can minimize risk subject to a constraint imposing a lower bound on expected return. Alternatively, expected return and risk can be combined in an objective function, to give the model:&lt;/p&gt; &lt;span&gt; &lt;p class=&#34;equation&#34;&gt;\[\left.\begin{array}{rrcl} \max &amp;amp; f(x) = 20x_1 + 16x_2 &amp;#8722; \theta[2x_1^2 + x_2^2 + (x_1 + x_2)^2] \\ \text {s.t.:} &amp;amp; &amp;amp; &amp;amp; \\ &amp;amp; x_1 + x_2 \leq 5 \\ &amp;amp; x_1, x_2 \geq 0 \\ &amp;amp; 0 \leq x_i \leq 5, i \in \{1,2\} \end{array}\right\}\]&lt;/p&gt; &lt;/span&gt; &lt;p&gt;Suppose \(\theta=1\). To two decimals, what is the objective value?&lt;/p&gt; &lt;p&gt;You can solve using spreadsheet optimization or using Julia. If you use Julia, you will need the following additional syntax: &lt;pre&gt;&lt;code&gt; Pkg.add(&#34;NLopt&#34;) yourModelVariable=Model(solver=NLoptSolver(algorithm=:LD_MMA)) @NLobjective(yourModelVariable, Max/Min, Function) @NLconstraint(yourModelVariable, Inequality)) &lt;/code&gt;&lt;/pre&gt;&lt;/p&gt; &lt;div class=&#34;wrapper-problem-response&#34; tabindex=&#34;-1&#34; aria-label=&#34;Question 1&#34; role=&#34;group&#34;&gt;&lt;div id=&#34;formulaequationinput_8706ba6e7738497194ac73239f763fee_2_1&#34; class=&#34;inputtype formulaequationinput&#34;&gt; &lt;div class=&#34;unanswered&#34;&gt; &lt;input type=&#34;text&#34; name=&#34;input_8706ba6e7738497194ac73239f763fee_2_1&#34; id=&#34;input_8706ba6e7738497194ac73239f763fee_2_1&#34; data-input-id=&#34;8706ba6e7738497194ac73239f763fee_2_1&#34; value=&#34;&#34; aria-describedby=&#34;status_8706ba6e7738497194ac73239f763fee_2_1&#34; size=&#34;20&#34;/&gt; &lt;span class=&#34;trailing_text&#34; id=&#34;trailing_text_8706ba6e7738497194ac73239f763fee_2_1&#34;/&gt; &lt;span class=&#34;status unanswered&#34; id=&#34;status_8706ba6e7738497194ac73239f763fee_2_1&#34; data-tooltip=&#34;Not yet answered.&#34;&gt; &lt;span class=&#34;sr&#34;&gt;unanswered&lt;/span&gt;&lt;span class=&#34;status-icon&#34; aria-hidden=&#34;true&#34;/&gt; &lt;/span&gt; &lt;p id=&#34;answer_8706ba6e7738497194ac73239f763fee_2_1&#34; class=&#34;answer&#34;/&gt; &lt;div id=&#34;input_8706ba6e7738497194ac73239f763fee_2_1_preview&#34; class=&#34;equation&#34;&gt; \(\) &lt;img src=&#34;/static/images/spinner.bc34f953403f.gif&#34; class=&#34;loading&#34; alt=&#34;Loading&#34;/&gt; &lt;/div&gt; &lt;/div&gt; &lt;div class=&#34;script_placeholder&#34; data-src=&#34;/static/js/capa/src/formula_equation_preview.b1967ab28c31.js&#34;/&gt; &lt;/div&gt;&lt;/div&gt; &lt;div class=&#34;solution-span&#34;&gt; &lt;span id=&#34;solution_8706ba6e7738497194ac73239f763fee_solution_1&#34;/&gt; &lt;/div&gt;&lt;/div&gt; &lt;div class=&#34;action&#34;&gt; &lt;input type=&#34;hidden&#34; name=&#34;problem_id&#34; value=&#34;PART A&#34; /&gt; &lt;div class=&#34;submit-attempt-container&#34;&gt; &lt;button type=&#34;button&#34; class=&#34;submit btn-brand&#34; data-submitting=&#34;Submitting&#34; data-value=&#34;Submit&#34; data-should-enable-submit-button=&#34;True&#34; aria-describedby=&#34;submission_feedback_8706ba6e7738497194ac73239f763fee&#34; &gt; &lt;span class=&#34;submit-label&#34;&gt;Submit&lt;/span&gt; &lt;/button&gt; &lt;div class=&#34;submission-feedback&#34; id=&#34;submission_feedback_8706ba6e7738497194ac73239f763fee&#34;&gt; &lt;span class=&#34;sr&#34;&gt;Some problems have options such as save, reset, hints, or show answer. These options follow the Submit button.&lt;/span&gt; &lt;/div&gt; &lt;/div&gt; &lt;div class=&#34;problem-action-buttons-wrapper&#34;&gt; &lt;/div&gt; &lt;/div&gt; &lt;div class=&#34;notification warning notification-gentle-alert is-hidden&#34; tabindex=&#34;-1&#34;&gt; &lt;span class=&#34;icon fa fa-exclamation-circle&#34; aria-hidden=&#34;true&#34;&gt;&lt;/span&gt; &lt;span class=&#34;notification-message&#34; aria-describedby=&#34;8706ba6e7738497194ac73239f763fee-problem-title&#34;&gt; &lt;/span&gt; &lt;div class=&#34;notification-btn-wrapper&#34;&gt; &lt;button type=&#34;button&#34; class=&#34;btn btn-default btn-small notification-btn review-btn sr&#34;&gt;Review&lt;/button&gt; &lt;/div&gt; &lt;/div&gt; &lt;div class=&#34;notification warning notification-save is-hidden&#34; tabindex=&#34;-1&#34;&gt; &lt;span class=&#34;icon fa fa-save&#34; aria-hidden=&#34;true&#34;&gt;&lt;/span&gt; &lt;span class=&#34;notification-message&#34; aria-describedby=&#34;8706ba6e7738497194ac73239f763fee-problem-title&#34;&gt;None &lt;/span&gt; &lt;div class=&#34;notification-btn-wrapper&#34;&gt; &lt;button type=&#34;button&#34; class=&#34;btn btn-default btn-small notification-btn review-btn sr&#34;&gt;Review&lt;/button&gt; &lt;/div&gt; &lt;/div&gt; &lt;div class=&#34;notification general notification-show-answer is-hidden&#34; tabindex=&#34;-1&#34;&gt; &lt;span class=&#34;icon fa fa-info-circle&#34; aria-hidden=&#34;true&#34;&gt;&lt;/span&gt; &lt;span class=&#34;notification-message&#34; aria-describedby=&#34;8706ba6e7738497194ac73239f763fee-problem-title&#34;&gt;Answers are displayed within the problem &lt;/span&gt; &lt;div class=&#34;notification-btn-wrapper&#34;&gt; &lt;button type=&#34;button&#34; class=&#34;btn btn-default btn-small notification-btn review-btn sr&#34;&gt;Review&lt;/button&gt; &lt;/div&gt; &lt;/div&gt; &lt;/div&gt; " data-graded="False"> <p class="loading-spinner"> <i class="fa fa-spinner fa-pulse fa-2x fa-fw"></i> <span class="sr">Loading&hellip;</span> </p> </div> </div> </div> <div class="vert vert-1" data-id="block-v1:MITx+15.053x+3T2016+type@problem+block@a4810e7372bf4d8dace6487fa9d2f052"> <div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-usage-id="block-v1:MITx+15.053x+3T2016+type@problem+block@a4810e7372bf4d8dace6487fa9d2f052" data-block-type="problem" data-graded="False" data-course-id="course-v1:MITx+15.053x+3T2016" data-has-score="True" data-request-token="6a146286852711ef8f57023e4242adeb" data-init="XBlockToXModuleShim" data-runtime-class="LmsRuntime" data-runtime-version="1"> <script type="json/xblock-args" class="xblock-json-init-args"> {"xmodule-type": "Problem"} </script> <div id="problem_a4810e7372bf4d8dace6487fa9d2f052" class="problems-wrapper" role="group" aria-labelledby="a4810e7372bf4d8dace6487fa9d2f052-problem-title" data-problem-id="block-v1:MITx+15.053x+3T2016+type@problem+block@a4810e7372bf4d8dace6487fa9d2f052" data-url="/courses/course-v1:MITx+15.053x+3T2016/xblock/block-v1:MITx+15.053x+3T2016+type@problem+block@a4810e7372bf4d8dace6487fa9d2f052/handler/xmodule_handler" data-problem-score="0" data-problem-total-possible="0" data-attempts-used="0" data-content=" &lt;h3 class=&#34;hd hd-3 problem-header&#34; id=&#34;a4810e7372bf4d8dace6487fa9d2f052-problem-title&#34; aria-describedby=&#34;block-v1:MITx+15.053x+3T2016+type@problem+block@a4810e7372bf4d8dace6487fa9d2f052-problem-progress&#34; tabindex=&#34;-1&#34;&gt; PART B &lt;/h3&gt; &lt;div class=&#34;problem-progress&#34; id=&#34;block-v1:MITx+15.053x+3T2016+type@problem+block@a4810e7372bf4d8dace6487fa9d2f052-problem-progress&#34;&gt;&lt;/div&gt; &lt;div class=&#34;problem&#34;&gt; &lt;div&gt; &lt;p&gt;Next, we would like to study the objective for different values of \(\theta\). Specifically, for \(\theta \in \{0, 1, \ldots, 5\} \), we would like to determine where the objective is maximized. &lt;/p&gt; &lt;p&gt;For which of these six values value of \(\theta\) is the optimum objective value maximized?&lt;/p&gt; &lt;p&gt;If you use Julia and JuMP, then create a vector containing the values of \(\theta\), and then use a loop. If you use a spreadsheet, solve the problem six times and record your answers.&lt;/p&gt; &lt;div class=&#34;wrapper-problem-response&#34; tabindex=&#34;-1&#34; aria-label=&#34;Question 1&#34; role=&#34;group&#34;&gt;&lt;div id=&#34;formulaequationinput_a4810e7372bf4d8dace6487fa9d2f052_2_1&#34; class=&#34;inputtype formulaequationinput&#34;&gt; &lt;div class=&#34;unanswered&#34;&gt; &lt;input type=&#34;text&#34; name=&#34;input_a4810e7372bf4d8dace6487fa9d2f052_2_1&#34; id=&#34;input_a4810e7372bf4d8dace6487fa9d2f052_2_1&#34; data-input-id=&#34;a4810e7372bf4d8dace6487fa9d2f052_2_1&#34; value=&#34;&#34; aria-describedby=&#34;status_a4810e7372bf4d8dace6487fa9d2f052_2_1&#34; size=&#34;20&#34;/&gt; &lt;span class=&#34;trailing_text&#34; id=&#34;trailing_text_a4810e7372bf4d8dace6487fa9d2f052_2_1&#34;/&gt; &lt;span class=&#34;status unanswered&#34; id=&#34;status_a4810e7372bf4d8dace6487fa9d2f052_2_1&#34; data-tooltip=&#34;Not yet answered.&#34;&gt; &lt;span class=&#34;sr&#34;&gt;unanswered&lt;/span&gt;&lt;span class=&#34;status-icon&#34; aria-hidden=&#34;true&#34;/&gt; &lt;/span&gt; &lt;p id=&#34;answer_a4810e7372bf4d8dace6487fa9d2f052_2_1&#34; class=&#34;answer&#34;/&gt; &lt;div id=&#34;input_a4810e7372bf4d8dace6487fa9d2f052_2_1_preview&#34; class=&#34;equation&#34;&gt; \(\) &lt;img src=&#34;/static/images/spinner.bc34f953403f.gif&#34; class=&#34;loading&#34; alt=&#34;Loading&#34;/&gt; &lt;/div&gt; &lt;/div&gt; &lt;div class=&#34;script_placeholder&#34; data-src=&#34;/static/js/capa/src/formula_equation_preview.b1967ab28c31.js&#34;/&gt; &lt;/div&gt;&lt;/div&gt; &lt;div class=&#34;solution-span&#34;&gt; &lt;span id=&#34;solution_a4810e7372bf4d8dace6487fa9d2f052_solution_1&#34;/&gt; &lt;/div&gt;&lt;/div&gt; &lt;div class=&#34;action&#34;&gt; &lt;input type=&#34;hidden&#34; name=&#34;problem_id&#34; value=&#34;PART B&#34; /&gt; &lt;div class=&#34;submit-attempt-container&#34;&gt; &lt;button type=&#34;button&#34; class=&#34;submit btn-brand&#34; data-submitting=&#34;Submitting&#34; data-value=&#34;Submit&#34; data-should-enable-submit-button=&#34;True&#34; aria-describedby=&#34;submission_feedback_a4810e7372bf4d8dace6487fa9d2f052&#34; &gt; &lt;span class=&#34;submit-label&#34;&gt;Submit&lt;/span&gt; &lt;/button&gt; &lt;div class=&#34;submission-feedback&#34; id=&#34;submission_feedback_a4810e7372bf4d8dace6487fa9d2f052&#34;&gt; &lt;span class=&#34;sr&#34;&gt;Some problems have options such as save, reset, hints, or show answer. These options follow the Submit button.&lt;/span&gt; &lt;/div&gt; &lt;/div&gt; &lt;div class=&#34;problem-action-buttons-wrapper&#34;&gt; &lt;/div&gt; &lt;/div&gt; &lt;div class=&#34;notification warning notification-gentle-alert is-hidden&#34; tabindex=&#34;-1&#34;&gt; &lt;span class=&#34;icon fa fa-exclamation-circle&#34; aria-hidden=&#34;true&#34;&gt;&lt;/span&gt; &lt;span class=&#34;notification-message&#34; aria-describedby=&#34;a4810e7372bf4d8dace6487fa9d2f052-problem-title&#34;&gt; &lt;/span&gt; &lt;div class=&#34;notification-btn-wrapper&#34;&gt; &lt;button type=&#34;button&#34; class=&#34;btn btn-default btn-small notification-btn review-btn sr&#34;&gt;Review&lt;/button&gt; &lt;/div&gt; &lt;/div&gt; &lt;div class=&#34;notification warning notification-save is-hidden&#34; tabindex=&#34;-1&#34;&gt; &lt;span class=&#34;icon fa fa-save&#34; aria-hidden=&#34;true&#34;&gt;&lt;/span&gt; &lt;span class=&#34;notification-message&#34; aria-describedby=&#34;a4810e7372bf4d8dace6487fa9d2f052-problem-title&#34;&gt;None &lt;/span&gt; &lt;div class=&#34;notification-btn-wrapper&#34;&gt; &lt;button type=&#34;button&#34; class=&#34;btn btn-default btn-small notification-btn review-btn sr&#34;&gt;Review&lt;/button&gt; &lt;/div&gt; &lt;/div&gt; &lt;div class=&#34;notification general notification-show-answer is-hidden&#34; tabindex=&#34;-1&#34;&gt; &lt;span class=&#34;icon fa fa-info-circle&#34; aria-hidden=&#34;true&#34;&gt;&lt;/span&gt; &lt;span class=&#34;notification-message&#34; aria-describedby=&#34;a4810e7372bf4d8dace6487fa9d2f052-problem-title&#34;&gt;Answers are displayed within the problem &lt;/span&gt; &lt;div class=&#34;notification-btn-wrapper&#34;&gt; &lt;button type=&#34;button&#34; class=&#34;btn btn-default btn-small notification-btn review-btn sr&#34;&gt;Review&lt;/button&gt; &lt;/div&gt; &lt;/div&gt; &lt;/div&gt; " data-graded="False"> <p class="loading-spinner"> <i class="fa fa-spinner fa-pulse fa-2x fa-fw"></i> <span class="sr">Loading&hellip;</span> </p> </div> </div> </div> <div class="vert vert-2" data-id="block-v1:MITx+15.053x+3T2016+type@problem+block@39dce5977471457bbdcbd3fbd72f6466"> <div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-usage-id="block-v1:MITx+15.053x+3T2016+type@problem+block@39dce5977471457bbdcbd3fbd72f6466" data-block-type="problem" data-graded="False" data-course-id="course-v1:MITx+15.053x+3T2016" data-has-score="True" data-request-token="6a146286852711ef8f57023e4242adeb" data-init="XBlockToXModuleShim" data-runtime-class="LmsRuntime" data-runtime-version="1"> <script type="json/xblock-args" class="xblock-json-init-args"> {"xmodule-type": "Problem"} </script> <div id="problem_39dce5977471457bbdcbd3fbd72f6466" class="problems-wrapper" role="group" aria-labelledby="39dce5977471457bbdcbd3fbd72f6466-problem-title" data-problem-id="block-v1:MITx+15.053x+3T2016+type@problem+block@39dce5977471457bbdcbd3fbd72f6466" data-url="/courses/course-v1:MITx+15.053x+3T2016/xblock/block-v1:MITx+15.053x+3T2016+type@problem+block@39dce5977471457bbdcbd3fbd72f6466/handler/xmodule_handler" data-problem-score="0" data-problem-total-possible="0" data-attempts-used="0" data-content=" &lt;h3 class=&#34;hd hd-3 problem-header&#34; id=&#34;39dce5977471457bbdcbd3fbd72f6466-problem-title&#34; aria-describedby=&#34;block-v1:MITx+15.053x+3T2016+type@problem+block@39dce5977471457bbdcbd3fbd72f6466-problem-progress&#34; tabindex=&#34;-1&#34;&gt; PART C &lt;/h3&gt; &lt;div class=&#34;problem-progress&#34; id=&#34;block-v1:MITx+15.053x+3T2016+type@problem+block@39dce5977471457bbdcbd3fbd72f6466-problem-progress&#34;&gt;&lt;/div&gt; &lt;div class=&#34;problem&#34;&gt; &lt;div&gt; &lt;p&gt;Next, we would like to plot the expected return as a function of \(\theta\) for \(\theta \in \{0, \ldots, 5\} \) .&lt;/p&gt; &lt;p&gt; &lt;i&gt;If you use Julia and JuMP, you will need the following additional syntax&lt;/i&gt; &lt;pre&gt;&lt;code&gt; Pkg.add(&#34;PyPlot&#34;) using PyPlot plot(xVector, yVector, label=&#34;your label&#34;, xlabel=&#34;...&#34;, ylabel=&#34;...&#34;) &lt;/code&gt;&lt;/pre&gt; &lt;/p&gt; &lt;div class=&#34;wrapper-problem-response&#34; tabindex=&#34;-1&#34; aria-label=&#34;Question 1&#34; role=&#34;group&#34;&gt;&lt;div class=&#34;choicegroup capa_inputtype&#34; id=&#34;inputtype_39dce5977471457bbdcbd3fbd72f6466_2_1&#34;&gt; &lt;fieldset aria-describedby=&#34;status_39dce5977471457bbdcbd3fbd72f6466_2_1&#34;&gt; &lt;div class=&#34;field&#34;&gt; &lt;input type=&#34;radio&#34; name=&#34;input_39dce5977471457bbdcbd3fbd72f6466_2_1&#34; id=&#34;input_39dce5977471457bbdcbd3fbd72f6466_2_1_choice_0&#34; class=&#34;field-input input-radio&#34; value=&#34;choice_0&#34;/&gt;&lt;label id=&#34;39dce5977471457bbdcbd3fbd72f6466_2_1-choice_0-label&#34; for=&#34;input_39dce5977471457bbdcbd3fbd72f6466_2_1_choice_0&#34; class=&#34;response-label field-label label-inline&#34; aria-describedby=&#34;status_39dce5977471457bbdcbd3fbd72f6466_2_1&#34;&gt; I am ready to have the answer shown. &lt;/label&gt; &lt;/div&gt; &lt;span id=&#34;answer_39dce5977471457bbdcbd3fbd72f6466_2_1&#34;/&gt; &lt;/fieldset&gt; &lt;div class=&#34;indicator-container&#34;&gt; &lt;span class=&#34;status unanswered&#34; id=&#34;status_39dce5977471457bbdcbd3fbd72f6466_2_1&#34; data-tooltip=&#34;Not yet answered.&#34;&gt; &lt;span class=&#34;sr&#34;&gt;unanswered&lt;/span&gt;&lt;span class=&#34;status-icon&#34; aria-hidden=&#34;true&#34;/&gt; &lt;/span&gt; &lt;/div&gt; &lt;/div&gt;&lt;/div&gt; &lt;div class=&#34;solution-span&#34;&gt; &lt;span id=&#34;solution_39dce5977471457bbdcbd3fbd72f6466_solution_1&#34;/&gt; &lt;/div&gt;&lt;/div&gt; &lt;div class=&#34;action&#34;&gt; &lt;input type=&#34;hidden&#34; name=&#34;problem_id&#34; value=&#34;PART C&#34; /&gt; &lt;div class=&#34;submit-attempt-container&#34;&gt; &lt;button type=&#34;button&#34; class=&#34;submit btn-brand&#34; data-submitting=&#34;Submitting&#34; data-value=&#34;Submit&#34; data-should-enable-submit-button=&#34;True&#34; aria-describedby=&#34;submission_feedback_39dce5977471457bbdcbd3fbd72f6466&#34; &gt; &lt;span class=&#34;submit-label&#34;&gt;Submit&lt;/span&gt; &lt;/button&gt; &lt;div class=&#34;submission-feedback&#34; id=&#34;submission_feedback_39dce5977471457bbdcbd3fbd72f6466&#34;&gt; &lt;span class=&#34;sr&#34;&gt;Some problems have options such as save, reset, hints, or show answer. These options follow the Submit button.&lt;/span&gt; &lt;/div&gt; &lt;/div&gt; &lt;div class=&#34;problem-action-buttons-wrapper&#34;&gt; &lt;/div&gt; &lt;/div&gt; &lt;div class=&#34;notification warning notification-gentle-alert is-hidden&#34; tabindex=&#34;-1&#34;&gt; &lt;span class=&#34;icon fa fa-exclamation-circle&#34; aria-hidden=&#34;true&#34;&gt;&lt;/span&gt; &lt;span class=&#34;notification-message&#34; aria-describedby=&#34;39dce5977471457bbdcbd3fbd72f6466-problem-title&#34;&gt; &lt;/span&gt; &lt;div class=&#34;notification-btn-wrapper&#34;&gt; &lt;button type=&#34;button&#34; class=&#34;btn btn-default btn-small notification-btn review-btn sr&#34;&gt;Review&lt;/button&gt; &lt;/div&gt; &lt;/div&gt; &lt;div class=&#34;notification warning notification-save is-hidden&#34; tabindex=&#34;-1&#34;&gt; &lt;span class=&#34;icon fa fa-save&#34; aria-hidden=&#34;true&#34;&gt;&lt;/span&gt; &lt;span class=&#34;notification-message&#34; aria-describedby=&#34;39dce5977471457bbdcbd3fbd72f6466-problem-title&#34;&gt;None &lt;/span&gt; &lt;div class=&#34;notification-btn-wrapper&#34;&gt; &lt;button type=&#34;button&#34; class=&#34;btn btn-default btn-small notification-btn review-btn sr&#34;&gt;Review&lt;/button&gt; &lt;/div&gt; &lt;/div&gt; &lt;div class=&#34;notification general notification-show-answer is-hidden&#34; tabindex=&#34;-1&#34;&gt; &lt;span class=&#34;icon fa fa-info-circle&#34; aria-hidden=&#34;true&#34;&gt;&lt;/span&gt; &lt;span class=&#34;notification-message&#34; aria-describedby=&#34;39dce5977471457bbdcbd3fbd72f6466-problem-title&#34;&gt;Answers are displayed within the problem &lt;/span&gt; &lt;div class=&#34;notification-btn-wrapper&#34;&gt; &lt;button type=&#34;button&#34; class=&#34;btn btn-default btn-small notification-btn review-btn sr&#34;&gt;Review&lt;/button&gt; &lt;/div&gt; &lt;/div&gt; &lt;/div&gt; " data-graded="False"> <p class="loading-spinner"> <i class="fa fa-spinner fa-pulse fa-2x fa-fw"></i> <span class="sr">Loading&hellip;</span> </p> </div> </div> </div> </div> </div>

<div class="xblock xblock-public_view xblock-public_view-vertical" data-usage-id="block-v1:MITx+15.053x+3T2016+type@vertical+block@c13c666983e5462d919380c1c34ba6c9" data-block-type="vertical" data-graded="False" data-course-id="course-v1:MITx+15.053x+3T2016" data-has-score="False" data-request-token="6a146286852711ef8f57023e4242adeb" data-init="VerticalStudentView" data-runtime-class="LmsRuntime" data-runtime-version="1"> <h2 class="hd hd-2 unit-title">Problem 3</h2> <div class="vert-mod"> <div class="vert vert-0" data-id="block-v1:MITx+15.053x+3T2016+type@problem+block@b181a7bf530f469f985ef118d2852e3f"> <div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-usage-id="block-v1:MITx+15.053x+3T2016+type@problem+block@b181a7bf530f469f985ef118d2852e3f" data-block-type="problem" data-graded="False" data-course-id="course-v1:MITx+15.053x+3T2016" data-has-score="True" data-request-token="6a146286852711ef8f57023e4242adeb" data-init="XBlockToXModuleShim" data-runtime-class="LmsRuntime" data-runtime-version="1"> <script type="json/xblock-args" class="xblock-json-init-args"> {"xmodule-type": "Problem"} </script> <div id="problem_b181a7bf530f469f985ef118d2852e3f" class="problems-wrapper" role="group" aria-labelledby="b181a7bf530f469f985ef118d2852e3f-problem-title" data-problem-id="block-v1:MITx+15.053x+3T2016+type@problem+block@b181a7bf530f469f985ef118d2852e3f" data-url="/courses/course-v1:MITx+15.053x+3T2016/xblock/block-v1:MITx+15.053x+3T2016+type@problem+block@b181a7bf530f469f985ef118d2852e3f/handler/xmodule_handler" data-problem-score="0" data-problem-total-possible="0" data-attempts-used="0" data-content=" &lt;h3 class=&#34;hd hd-3 problem-header&#34; id=&#34;b181a7bf530f469f985ef118d2852e3f-problem-title&#34; aria-describedby=&#34;block-v1:MITx+15.053x+3T2016+type@problem+block@b181a7bf530f469f985ef118d2852e3f-problem-progress&#34; tabindex=&#34;-1&#34;&gt; PART A &lt;/h3&gt; &lt;div class=&#34;problem-progress&#34; id=&#34;block-v1:MITx+15.053x+3T2016+type@problem+block@b181a7bf530f469f985ef118d2852e3f-problem-progress&#34;&gt;&lt;/div&gt; &lt;div class=&#34;problem&#34;&gt; &lt;div&gt; &lt;p&gt;Consider the following second-order cone programming problem:&lt;/p&gt; &lt;span&gt; &lt;p class=&#34;equation&#34;&gt;\[\left.\begin{array}{rrcl} \min &amp;amp; t \\ \text {s.t.:} &amp;amp; &amp;amp; &amp;amp; \\ &amp;amp; ||x||_2 \leq t \\ &amp;amp; x_1+x_2 \geq 1 \\ &amp;amp; 0 \leq x_i \leq 1, i \in \{1,2\} \end{array}\right\}\]&lt;/p&gt; &lt;/span&gt; &lt;p&gt;You can solve using spreadsheet optimization or using Julia. If you use Julia, you will need the following additional syntax: &lt;pre&gt;&lt;code&gt; Pkg.add(&#34;ECOS&#34;) using ECOS @constraint(yourModelVariable, soc,norm2{x[i], i=start:end} [insert inequality] t) yourModelVariable=Model(solver=ECOSSolver()) @show getvalue(variable of interest) &lt;/code&gt;&lt;/pre&gt;&lt;/p&gt; &lt;p&gt;To two decimals, what is the optimal objective value?&lt;/p&gt; &lt;div class=&#34;wrapper-problem-response&#34; tabindex=&#34;-1&#34; aria-label=&#34;Question 1&#34; role=&#34;group&#34;&gt;&lt;div id=&#34;formulaequationinput_b181a7bf530f469f985ef118d2852e3f_2_1&#34; class=&#34;inputtype formulaequationinput&#34;&gt; &lt;div class=&#34;unanswered&#34;&gt; &lt;input type=&#34;text&#34; name=&#34;input_b181a7bf530f469f985ef118d2852e3f_2_1&#34; id=&#34;input_b181a7bf530f469f985ef118d2852e3f_2_1&#34; data-input-id=&#34;b181a7bf530f469f985ef118d2852e3f_2_1&#34; value=&#34;&#34; aria-describedby=&#34;status_b181a7bf530f469f985ef118d2852e3f_2_1&#34; size=&#34;20&#34;/&gt; &lt;span class=&#34;trailing_text&#34; id=&#34;trailing_text_b181a7bf530f469f985ef118d2852e3f_2_1&#34;/&gt; &lt;span class=&#34;status unanswered&#34; id=&#34;status_b181a7bf530f469f985ef118d2852e3f_2_1&#34; data-tooltip=&#34;Not yet answered.&#34;&gt; &lt;span class=&#34;sr&#34;&gt;unanswered&lt;/span&gt;&lt;span class=&#34;status-icon&#34; aria-hidden=&#34;true&#34;/&gt; &lt;/span&gt; &lt;p id=&#34;answer_b181a7bf530f469f985ef118d2852e3f_2_1&#34; class=&#34;answer&#34;/&gt; &lt;div id=&#34;input_b181a7bf530f469f985ef118d2852e3f_2_1_preview&#34; class=&#34;equation&#34;&gt; \(\) &lt;img src=&#34;/static/images/spinner.bc34f953403f.gif&#34; class=&#34;loading&#34; alt=&#34;Loading&#34;/&gt; &lt;/div&gt; &lt;/div&gt; &lt;div class=&#34;script_placeholder&#34; data-src=&#34;/static/js/capa/src/formula_equation_preview.b1967ab28c31.js&#34;/&gt; &lt;/div&gt;&lt;/div&gt; &lt;div class=&#34;solution-span&#34;&gt; &lt;span id=&#34;solution_b181a7bf530f469f985ef118d2852e3f_solution_1&#34;/&gt; &lt;/div&gt;&lt;/div&gt; &lt;div class=&#34;action&#34;&gt; &lt;input type=&#34;hidden&#34; name=&#34;problem_id&#34; value=&#34;PART A&#34; /&gt; &lt;div class=&#34;submit-attempt-container&#34;&gt; &lt;button type=&#34;button&#34; class=&#34;submit btn-brand&#34; data-submitting=&#34;Submitting&#34; data-value=&#34;Submit&#34; data-should-enable-submit-button=&#34;True&#34; aria-describedby=&#34;submission_feedback_b181a7bf530f469f985ef118d2852e3f&#34; &gt; &lt;span class=&#34;submit-label&#34;&gt;Submit&lt;/span&gt; &lt;/button&gt; &lt;div class=&#34;submission-feedback&#34; id=&#34;submission_feedback_b181a7bf530f469f985ef118d2852e3f&#34;&gt; &lt;span class=&#34;sr&#34;&gt;Some problems have options such as save, reset, hints, or show answer. These options follow the Submit button.&lt;/span&gt; &lt;/div&gt; &lt;/div&gt; &lt;div class=&#34;problem-action-buttons-wrapper&#34;&gt; &lt;/div&gt; &lt;/div&gt; &lt;div class=&#34;notification warning notification-gentle-alert is-hidden&#34; tabindex=&#34;-1&#34;&gt; &lt;span class=&#34;icon fa fa-exclamation-circle&#34; aria-hidden=&#34;true&#34;&gt;&lt;/span&gt; &lt;span class=&#34;notification-message&#34; aria-describedby=&#34;b181a7bf530f469f985ef118d2852e3f-problem-title&#34;&gt; &lt;/span&gt; &lt;div class=&#34;notification-btn-wrapper&#34;&gt; &lt;button type=&#34;button&#34; class=&#34;btn btn-default btn-small notification-btn review-btn sr&#34;&gt;Review&lt;/button&gt; &lt;/div&gt; &lt;/div&gt; &lt;div class=&#34;notification warning notification-save is-hidden&#34; tabindex=&#34;-1&#34;&gt; &lt;span class=&#34;icon fa fa-save&#34; aria-hidden=&#34;true&#34;&gt;&lt;/span&gt; &lt;span class=&#34;notification-message&#34; aria-describedby=&#34;b181a7bf530f469f985ef118d2852e3f-problem-title&#34;&gt;None &lt;/span&gt; &lt;div class=&#34;notification-btn-wrapper&#34;&gt; &lt;button type=&#34;button&#34; class=&#34;btn btn-default btn-small notification-btn review-btn sr&#34;&gt;Review&lt;/button&gt; &lt;/div&gt; &lt;/div&gt; &lt;div class=&#34;notification general notification-show-answer is-hidden&#34; tabindex=&#34;-1&#34;&gt; &lt;span class=&#34;icon fa fa-info-circle&#34; aria-hidden=&#34;true&#34;&gt;&lt;/span&gt; &lt;span class=&#34;notification-message&#34; aria-describedby=&#34;b181a7bf530f469f985ef118d2852e3f-problem-title&#34;&gt;Answers are displayed within the problem &lt;/span&gt; &lt;div class=&#34;notification-btn-wrapper&#34;&gt; &lt;button type=&#34;button&#34; class=&#34;btn btn-default btn-small notification-btn review-btn sr&#34;&gt;Review&lt;/button&gt; &lt;/div&gt; &lt;/div&gt; &lt;/div&gt; " data-graded="False"> <p class="loading-spinner"> <i class="fa fa-spinner fa-pulse fa-2x fa-fw"></i> <span class="sr">Loading&hellip;</span> </p> </div> </div> </div> </div> </div>

<div class="xblock xblock-public_view xblock-public_view-vertical" data-usage-id="block-v1:MITx+15.053x+3T2016+type@vertical+block@f3c5d83620a3469e81fe11996885afc1" data-block-type="vertical" data-graded="False" data-course-id="course-v1:MITx+15.053x+3T2016" data-has-score="False" data-request-token="6a146286852711ef8f57023e4242adeb" data-init="VerticalStudentView" data-runtime-class="LmsRuntime" data-runtime-version="1"> <h2 class="hd hd-2 unit-title">Problem 4</h2> <div class="vert-mod"> <div class="vert vert-0" data-id="block-v1:MITx+15.053x+3T2016+type@problem+block@cc5988ca34e04c0497eb23d6cab93319"> <div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-usage-id="block-v1:MITx+15.053x+3T2016+type@problem+block@cc5988ca34e04c0497eb23d6cab93319" data-block-type="problem" data-graded="False" data-course-id="course-v1:MITx+15.053x+3T2016" data-has-score="True" data-request-token="6a146286852711ef8f57023e4242adeb" data-init="XBlockToXModuleShim" data-runtime-class="LmsRuntime" data-runtime-version="1"> <script type="json/xblock-args" class="xblock-json-init-args"> {"xmodule-type": "Problem"} </script> <div id="problem_cc5988ca34e04c0497eb23d6cab93319" class="problems-wrapper" role="group" aria-labelledby="cc5988ca34e04c0497eb23d6cab93319-problem-title" data-problem-id="block-v1:MITx+15.053x+3T2016+type@problem+block@cc5988ca34e04c0497eb23d6cab93319" data-url="/courses/course-v1:MITx+15.053x+3T2016/xblock/block-v1:MITx+15.053x+3T2016+type@problem+block@cc5988ca34e04c0497eb23d6cab93319/handler/xmodule_handler" data-problem-score="0" data-problem-total-possible="0" data-attempts-used="0" data-content=" &lt;h3 class=&#34;hd hd-3 problem-header&#34; id=&#34;cc5988ca34e04c0497eb23d6cab93319-problem-title&#34; aria-describedby=&#34;block-v1:MITx+15.053x+3T2016+type@problem+block@cc5988ca34e04c0497eb23d6cab93319-problem-progress&#34; tabindex=&#34;-1&#34;&gt; PART A &lt;/h3&gt; &lt;div class=&#34;problem-progress&#34; id=&#34;block-v1:MITx+15.053x+3T2016+type@problem+block@cc5988ca34e04c0497eb23d6cab93319-problem-progress&#34;&gt;&lt;/div&gt; &lt;div class=&#34;problem&#34;&gt; &lt;div&gt; &lt;p&gt; For each part, determine whether the function is convex or not over its domain.&lt;/p&gt; &lt;p&gt; &lt;a class=&#34;modal-content&#34; href=&#34;/assets/courseware/v1/6010f620a9d683aaf03157cc45771391/asset-v1:MITx+15.053x+3T2016+type@asset+block/R6P4aa.png&#34;&gt; &lt;img width=&#34;600&#34; height=&#34;439&#34; alt=&#34;Plot1&#34; src=&#34;/assets/courseware/v1/6010f620a9d683aaf03157cc45771391/asset-v1:MITx+15.053x+3T2016+type@asset+block/R6P4aa.png&#34;/&gt; &lt;/a&gt; &lt;/p&gt; &lt;p&gt; \( f(x) = |x-1| \) for \(-3 \le x \le 3 \). &lt;/p&gt; &lt;div class=&#34;wrapper-problem-response&#34; tabindex=&#34;-1&#34; aria-label=&#34;Question 1&#34; role=&#34;group&#34;&gt;&lt;div class=&#34;choicegroup capa_inputtype&#34; id=&#34;inputtype_cc5988ca34e04c0497eb23d6cab93319_2_1&#34;&gt; &lt;fieldset aria-describedby=&#34;status_cc5988ca34e04c0497eb23d6cab93319_2_1&#34;&gt; &lt;div class=&#34;field&#34;&gt; &lt;input type=&#34;radio&#34; name=&#34;input_cc5988ca34e04c0497eb23d6cab93319_2_1&#34; id=&#34;input_cc5988ca34e04c0497eb23d6cab93319_2_1_choice_1&#34; class=&#34;field-input input-radio&#34; value=&#34;choice_1&#34;/&gt;&lt;label id=&#34;cc5988ca34e04c0497eb23d6cab93319_2_1-choice_1-label&#34; for=&#34;input_cc5988ca34e04c0497eb23d6cab93319_2_1_choice_1&#34; class=&#34;response-label field-label label-inline&#34; aria-describedby=&#34;status_cc5988ca34e04c0497eb23d6cab93319_2_1&#34;&gt; Not Convex &lt;/label&gt; &lt;/div&gt; &lt;div class=&#34;field&#34;&gt; &lt;input type=&#34;radio&#34; name=&#34;input_cc5988ca34e04c0497eb23d6cab93319_2_1&#34; id=&#34;input_cc5988ca34e04c0497eb23d6cab93319_2_1_choice_0&#34; class=&#34;field-input input-radio&#34; value=&#34;choice_0&#34;/&gt;&lt;label id=&#34;cc5988ca34e04c0497eb23d6cab93319_2_1-choice_0-label&#34; for=&#34;input_cc5988ca34e04c0497eb23d6cab93319_2_1_choice_0&#34; class=&#34;response-label field-label label-inline&#34; aria-describedby=&#34;status_cc5988ca34e04c0497eb23d6cab93319_2_1&#34;&gt; Convex &lt;/label&gt; &lt;/div&gt; &lt;span id=&#34;answer_cc5988ca34e04c0497eb23d6cab93319_2_1&#34;/&gt; &lt;/fieldset&gt; &lt;div class=&#34;indicator-container&#34;&gt; &lt;span class=&#34;status unanswered&#34; id=&#34;status_cc5988ca34e04c0497eb23d6cab93319_2_1&#34; data-tooltip=&#34;Not yet answered.&#34;&gt; &lt;span class=&#34;sr&#34;&gt;unanswered&lt;/span&gt;&lt;span class=&#34;status-icon&#34; aria-hidden=&#34;true&#34;/&gt; &lt;/span&gt; &lt;/div&gt; &lt;/div&gt;&lt;/div&gt; &lt;div class=&#34;solution-span&#34;&gt; &lt;span id=&#34;solution_cc5988ca34e04c0497eb23d6cab93319_solution_1&#34;/&gt; &lt;/div&gt;&lt;/div&gt; &lt;div class=&#34;action&#34;&gt; &lt;input type=&#34;hidden&#34; name=&#34;problem_id&#34; value=&#34;PART A&#34; /&gt; &lt;div class=&#34;submit-attempt-container&#34;&gt; &lt;button type=&#34;button&#34; class=&#34;submit btn-brand&#34; data-submitting=&#34;Submitting&#34; data-value=&#34;Submit&#34; data-should-enable-submit-button=&#34;True&#34; aria-describedby=&#34;submission_feedback_cc5988ca34e04c0497eb23d6cab93319&#34; &gt; &lt;span class=&#34;submit-label&#34;&gt;Submit&lt;/span&gt; &lt;/button&gt; &lt;div class=&#34;submission-feedback&#34; id=&#34;submission_feedback_cc5988ca34e04c0497eb23d6cab93319&#34;&gt; &lt;span class=&#34;sr&#34;&gt;Some problems have options such as save, reset, hints, or show answer. These options follow the Submit button.&lt;/span&gt; &lt;/div&gt; &lt;/div&gt; &lt;div class=&#34;problem-action-buttons-wrapper&#34;&gt; &lt;/div&gt; &lt;/div&gt; &lt;div class=&#34;notification warning notification-gentle-alert is-hidden&#34; tabindex=&#34;-1&#34;&gt; &lt;span class=&#34;icon fa fa-exclamation-circle&#34; aria-hidden=&#34;true&#34;&gt;&lt;/span&gt; &lt;span class=&#34;notification-message&#34; aria-describedby=&#34;cc5988ca34e04c0497eb23d6cab93319-problem-title&#34;&gt; &lt;/span&gt; &lt;div class=&#34;notification-btn-wrapper&#34;&gt; &lt;button type=&#34;button&#34; class=&#34;btn btn-default btn-small notification-btn review-btn sr&#34;&gt;Review&lt;/button&gt; &lt;/div&gt; &lt;/div&gt; &lt;div class=&#34;notification warning notification-save is-hidden&#34; tabindex=&#34;-1&#34;&gt; &lt;span class=&#34;icon fa fa-save&#34; aria-hidden=&#34;true&#34;&gt;&lt;/span&gt; &lt;span class=&#34;notification-message&#34; aria-describedby=&#34;cc5988ca34e04c0497eb23d6cab93319-problem-title&#34;&gt;None &lt;/span&gt; &lt;div class=&#34;notification-btn-wrapper&#34;&gt; &lt;button type=&#34;button&#34; class=&#34;btn btn-default btn-small notification-btn review-btn sr&#34;&gt;Review&lt;/button&gt; &lt;/div&gt; &lt;/div&gt; &lt;div class=&#34;notification general notification-show-answer is-hidden&#34; tabindex=&#34;-1&#34;&gt; &lt;span class=&#34;icon fa fa-info-circle&#34; aria-hidden=&#34;true&#34;&gt;&lt;/span&gt; &lt;span class=&#34;notification-message&#34; aria-describedby=&#34;cc5988ca34e04c0497eb23d6cab93319-problem-title&#34;&gt;Answers are displayed within the problem &lt;/span&gt; &lt;div class=&#34;notification-btn-wrapper&#34;&gt; &lt;button type=&#34;button&#34; class=&#34;btn btn-default btn-small notification-btn review-btn sr&#34;&gt;Review&lt;/button&gt; &lt;/div&gt; &lt;/div&gt; &lt;/div&gt; " data-graded="False"> <p class="loading-spinner"> <i class="fa fa-spinner fa-pulse fa-2x fa-fw"></i> <span class="sr">Loading&hellip;</span> </p> </div> </div> </div> <div class="vert vert-1" data-id="block-v1:MITx+15.053x+3T2016+type@problem+block@7e3798351edb4fe19daf03770db8feaf"> <div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-usage-id="block-v1:MITx+15.053x+3T2016+type@problem+block@7e3798351edb4fe19daf03770db8feaf" data-block-type="problem" data-graded="False" data-course-id="course-v1:MITx+15.053x+3T2016" data-has-score="True" data-request-token="6a146286852711ef8f57023e4242adeb" data-init="XBlockToXModuleShim" data-runtime-class="LmsRuntime" data-runtime-version="1"> <script type="json/xblock-args" class="xblock-json-init-args"> {"xmodule-type": "Problem"} </script> <div id="problem_7e3798351edb4fe19daf03770db8feaf" class="problems-wrapper" role="group" aria-labelledby="7e3798351edb4fe19daf03770db8feaf-problem-title" data-problem-id="block-v1:MITx+15.053x+3T2016+type@problem+block@7e3798351edb4fe19daf03770db8feaf" data-url="/courses/course-v1:MITx+15.053x+3T2016/xblock/block-v1:MITx+15.053x+3T2016+type@problem+block@7e3798351edb4fe19daf03770db8feaf/handler/xmodule_handler" data-problem-score="0" data-problem-total-possible="0" data-attempts-used="0" data-content=" &lt;h3 class=&#34;hd hd-3 problem-header&#34; id=&#34;7e3798351edb4fe19daf03770db8feaf-problem-title&#34; aria-describedby=&#34;block-v1:MITx+15.053x+3T2016+type@problem+block@7e3798351edb4fe19daf03770db8feaf-problem-progress&#34; tabindex=&#34;-1&#34;&gt; PART B &lt;/h3&gt; &lt;div class=&#34;problem-progress&#34; id=&#34;block-v1:MITx+15.053x+3T2016+type@problem+block@7e3798351edb4fe19daf03770db8feaf-problem-progress&#34;&gt;&lt;/div&gt; &lt;div class=&#34;problem&#34;&gt; &lt;div&gt; &lt;p&gt; &lt;a class=&#34;modal-content&#34; href=&#34;/assets/courseware/v1/4532915e2013410209e279903c0ca056/asset-v1:MITx+15.053x+3T2016+type@asset+block/R6P4bb.png&#34;&gt; &lt;img width=&#34;600&#34; height=&#34;439&#34; alt=&#34;Plot1&#34; src=&#34;/assets/courseware/v1/4532915e2013410209e279903c0ca056/asset-v1:MITx+15.053x+3T2016+type@asset+block/R6P4bb.png&#34;/&gt; &lt;/a&gt; &lt;/p&gt; &lt;p&gt; \( g(x) = x^2 -3 \) for \(-3 \le x \le 3 \). &lt;/p&gt; &lt;div class=&#34;wrapper-problem-response&#34; tabindex=&#34;-1&#34; aria-label=&#34;Question 1&#34; role=&#34;group&#34;&gt;&lt;div class=&#34;choicegroup capa_inputtype&#34; id=&#34;inputtype_7e3798351edb4fe19daf03770db8feaf_2_1&#34;&gt; &lt;fieldset aria-describedby=&#34;status_7e3798351edb4fe19daf03770db8feaf_2_1&#34;&gt; &lt;div class=&#34;field&#34;&gt; &lt;input type=&#34;radio&#34; name=&#34;input_7e3798351edb4fe19daf03770db8feaf_2_1&#34; id=&#34;input_7e3798351edb4fe19daf03770db8feaf_2_1_choice_1&#34; class=&#34;field-input input-radio&#34; value=&#34;choice_1&#34;/&gt;&lt;label id=&#34;7e3798351edb4fe19daf03770db8feaf_2_1-choice_1-label&#34; for=&#34;input_7e3798351edb4fe19daf03770db8feaf_2_1_choice_1&#34; class=&#34;response-label field-label label-inline&#34; aria-describedby=&#34;status_7e3798351edb4fe19daf03770db8feaf_2_1&#34;&gt; Not Convex &lt;/label&gt; &lt;/div&gt; &lt;div class=&#34;field&#34;&gt; &lt;input type=&#34;radio&#34; name=&#34;input_7e3798351edb4fe19daf03770db8feaf_2_1&#34; id=&#34;input_7e3798351edb4fe19daf03770db8feaf_2_1_choice_0&#34; class=&#34;field-input input-radio&#34; value=&#34;choice_0&#34;/&gt;&lt;label id=&#34;7e3798351edb4fe19daf03770db8feaf_2_1-choice_0-label&#34; for=&#34;input_7e3798351edb4fe19daf03770db8feaf_2_1_choice_0&#34; class=&#34;response-label field-label label-inline&#34; aria-describedby=&#34;status_7e3798351edb4fe19daf03770db8feaf_2_1&#34;&gt; Convex &lt;/label&gt; &lt;/div&gt; &lt;span id=&#34;answer_7e3798351edb4fe19daf03770db8feaf_2_1&#34;/&gt; &lt;/fieldset&gt; &lt;div class=&#34;indicator-container&#34;&gt; &lt;span class=&#34;status unanswered&#34; id=&#34;status_7e3798351edb4fe19daf03770db8feaf_2_1&#34; data-tooltip=&#34;Not yet answered.&#34;&gt; &lt;span class=&#34;sr&#34;&gt;unanswered&lt;/span&gt;&lt;span class=&#34;status-icon&#34; aria-hidden=&#34;true&#34;/&gt; &lt;/span&gt; &lt;/div&gt; &lt;/div&gt;&lt;/div&gt; &lt;div class=&#34;solution-span&#34;&gt; &lt;span id=&#34;solution_7e3798351edb4fe19daf03770db8feaf_solution_1&#34;/&gt; &lt;/div&gt;&lt;/div&gt; &lt;div class=&#34;action&#34;&gt; &lt;input type=&#34;hidden&#34; name=&#34;problem_id&#34; value=&#34;PART B&#34; /&gt; &lt;div class=&#34;submit-attempt-container&#34;&gt; &lt;button type=&#34;button&#34; class=&#34;submit btn-brand&#34; data-submitting=&#34;Submitting&#34; data-value=&#34;Submit&#34; data-should-enable-submit-button=&#34;True&#34; aria-describedby=&#34;submission_feedback_7e3798351edb4fe19daf03770db8feaf&#34; &gt; &lt;span class=&#34;submit-label&#34;&gt;Submit&lt;/span&gt; &lt;/button&gt; &lt;div class=&#34;submission-feedback&#34; id=&#34;submission_feedback_7e3798351edb4fe19daf03770db8feaf&#34;&gt; &lt;span class=&#34;sr&#34;&gt;Some problems have options such as save, reset, hints, or show answer. These options follow the Submit button.&lt;/span&gt; &lt;/div&gt; &lt;/div&gt; &lt;div class=&#34;problem-action-buttons-wrapper&#34;&gt; &lt;/div&gt; &lt;/div&gt; &lt;div class=&#34;notification warning notification-gentle-alert is-hidden&#34; tabindex=&#34;-1&#34;&gt; &lt;span class=&#34;icon fa fa-exclamation-circle&#34; aria-hidden=&#34;true&#34;&gt;&lt;/span&gt; &lt;span class=&#34;notification-message&#34; aria-describedby=&#34;7e3798351edb4fe19daf03770db8feaf-problem-title&#34;&gt; &lt;/span&gt; &lt;div class=&#34;notification-btn-wrapper&#34;&gt; &lt;button type=&#34;button&#34; class=&#34;btn btn-default btn-small notification-btn review-btn sr&#34;&gt;Review&lt;/button&gt; &lt;/div&gt; &lt;/div&gt; &lt;div class=&#34;notification warning notification-save is-hidden&#34; tabindex=&#34;-1&#34;&gt; &lt;span class=&#34;icon fa fa-save&#34; aria-hidden=&#34;true&#34;&gt;&lt;/span&gt; &lt;span class=&#34;notification-message&#34; aria-describedby=&#34;7e3798351edb4fe19daf03770db8feaf-problem-title&#34;&gt;None &lt;/span&gt; &lt;div class=&#34;notification-btn-wrapper&#34;&gt; &lt;button type=&#34;button&#34; class=&#34;btn btn-default btn-small notification-btn review-btn sr&#34;&gt;Review&lt;/button&gt; &lt;/div&gt; &lt;/div&gt; &lt;div class=&#34;notification general notification-show-answer is-hidden&#34; tabindex=&#34;-1&#34;&gt; &lt;span class=&#34;icon fa fa-info-circle&#34; aria-hidden=&#34;true&#34;&gt;&lt;/span&gt; &lt;span class=&#34;notification-message&#34; aria-describedby=&#34;7e3798351edb4fe19daf03770db8feaf-problem-title&#34;&gt;Answers are displayed within the problem &lt;/span&gt; &lt;div class=&#34;notification-btn-wrapper&#34;&gt; &lt;button type=&#34;button&#34; class=&#34;btn btn-default btn-small notification-btn review-btn sr&#34;&gt;Review&lt;/button&gt; &lt;/div&gt; &lt;/div&gt; &lt;/div&gt; " data-graded="False"> <p class="loading-spinner"> <i class="fa fa-spinner fa-pulse fa-2x fa-fw"></i> <span class="sr">Loading&hellip;</span> </p> </div> </div> </div> <div class="vert vert-2" data-id="block-v1:MITx+15.053x+3T2016+type@problem+block@2925dae3ab7d496ca438f506cae8baef"> <div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-usage-id="block-v1:MITx+15.053x+3T2016+type@problem+block@2925dae3ab7d496ca438f506cae8baef" data-block-type="problem" data-graded="False" data-course-id="course-v1:MITx+15.053x+3T2016" data-has-score="True" data-request-token="6a146286852711ef8f57023e4242adeb" data-init="XBlockToXModuleShim" data-runtime-class="LmsRuntime" data-runtime-version="1"> <script type="json/xblock-args" class="xblock-json-init-args"> {"xmodule-type": "Problem"} </script> <div id="problem_2925dae3ab7d496ca438f506cae8baef" class="problems-wrapper" role="group" aria-labelledby="2925dae3ab7d496ca438f506cae8baef-problem-title" data-problem-id="block-v1:MITx+15.053x+3T2016+type@problem+block@2925dae3ab7d496ca438f506cae8baef" data-url="/courses/course-v1:MITx+15.053x+3T2016/xblock/block-v1:MITx+15.053x+3T2016+type@problem+block@2925dae3ab7d496ca438f506cae8baef/handler/xmodule_handler" data-problem-score="0" data-problem-total-possible="0" data-attempts-used="0" data-content=" &lt;h3 class=&#34;hd hd-3 problem-header&#34; id=&#34;2925dae3ab7d496ca438f506cae8baef-problem-title&#34; aria-describedby=&#34;block-v1:MITx+15.053x+3T2016+type@problem+block@2925dae3ab7d496ca438f506cae8baef-problem-progress&#34; tabindex=&#34;-1&#34;&gt; PART C &lt;/h3&gt; &lt;div class=&#34;problem-progress&#34; id=&#34;block-v1:MITx+15.053x+3T2016+type@problem+block@2925dae3ab7d496ca438f506cae8baef-problem-progress&#34;&gt;&lt;/div&gt; &lt;div class=&#34;problem&#34;&gt; &lt;div&gt; &lt;p&gt; &lt;a class=&#34;modal-content&#34; href=&#34;/assets/courseware/v1/0af340703118e2ba9a1ff7106c027bf2/asset-v1:MITx+15.053x+3T2016+type@asset+block/R6P4cc.png&#34;&gt; &lt;img width=&#34;600&#34; height=&#34;439&#34; alt=&#34;Plot1&#34; src=&#34;/assets/courseware/v1/0af340703118e2ba9a1ff7106c027bf2/asset-v1:MITx+15.053x+3T2016+type@asset+block/R6P4cc.png&#34;/&gt; &lt;/a&gt; &lt;/p&gt; &lt;p&gt; \( h(x) = \max\{|x-1| , x^2-3\} \) for \( -3 \le x \le 3. \) &lt;/p&gt; &lt;div class=&#34;wrapper-problem-response&#34; tabindex=&#34;-1&#34; aria-label=&#34;Question 1&#34; role=&#34;group&#34;&gt;&lt;div class=&#34;choicegroup capa_inputtype&#34; id=&#34;inputtype_2925dae3ab7d496ca438f506cae8baef_2_1&#34;&gt; &lt;fieldset aria-describedby=&#34;status_2925dae3ab7d496ca438f506cae8baef_2_1&#34;&gt; &lt;div class=&#34;field&#34;&gt; &lt;input type=&#34;radio&#34; name=&#34;input_2925dae3ab7d496ca438f506cae8baef_2_1&#34; id=&#34;input_2925dae3ab7d496ca438f506cae8baef_2_1_choice_1&#34; class=&#34;field-input input-radio&#34; value=&#34;choice_1&#34;/&gt;&lt;label id=&#34;2925dae3ab7d496ca438f506cae8baef_2_1-choice_1-label&#34; for=&#34;input_2925dae3ab7d496ca438f506cae8baef_2_1_choice_1&#34; class=&#34;response-label field-label label-inline&#34; aria-describedby=&#34;status_2925dae3ab7d496ca438f506cae8baef_2_1&#34;&gt; Not Convex &lt;/label&gt; &lt;/div&gt; &lt;div class=&#34;field&#34;&gt; &lt;input type=&#34;radio&#34; name=&#34;input_2925dae3ab7d496ca438f506cae8baef_2_1&#34; id=&#34;input_2925dae3ab7d496ca438f506cae8baef_2_1_choice_0&#34; class=&#34;field-input input-radio&#34; value=&#34;choice_0&#34;/&gt;&lt;label id=&#34;2925dae3ab7d496ca438f506cae8baef_2_1-choice_0-label&#34; for=&#34;input_2925dae3ab7d496ca438f506cae8baef_2_1_choice_0&#34; class=&#34;response-label field-label label-inline&#34; aria-describedby=&#34;status_2925dae3ab7d496ca438f506cae8baef_2_1&#34;&gt; Convex &lt;/label&gt; &lt;/div&gt; &lt;span id=&#34;answer_2925dae3ab7d496ca438f506cae8baef_2_1&#34;/&gt; &lt;/fieldset&gt; &lt;div class=&#34;indicator-container&#34;&gt; &lt;span class=&#34;status unanswered&#34; id=&#34;status_2925dae3ab7d496ca438f506cae8baef_2_1&#34; data-tooltip=&#34;Not yet answered.&#34;&gt; &lt;span class=&#34;sr&#34;&gt;unanswered&lt;/span&gt;&lt;span class=&#34;status-icon&#34; aria-hidden=&#34;true&#34;/&gt; &lt;/span&gt; &lt;/div&gt; &lt;/div&gt;&lt;/div&gt; &lt;div class=&#34;solution-span&#34;&gt; &lt;span id=&#34;solution_2925dae3ab7d496ca438f506cae8baef_solution_1&#34;/&gt; &lt;/div&gt;&lt;/div&gt; &lt;div class=&#34;action&#34;&gt; &lt;input type=&#34;hidden&#34; name=&#34;problem_id&#34; value=&#34;PART C&#34; /&gt; &lt;div class=&#34;submit-attempt-container&#34;&gt; &lt;button type=&#34;button&#34; class=&#34;submit btn-brand&#34; data-submitting=&#34;Submitting&#34; data-value=&#34;Submit&#34; data-should-enable-submit-button=&#34;True&#34; aria-describedby=&#34;submission_feedback_2925dae3ab7d496ca438f506cae8baef&#34; &gt; &lt;span class=&#34;submit-label&#34;&gt;Submit&lt;/span&gt; &lt;/button&gt; &lt;div class=&#34;submission-feedback&#34; id=&#34;submission_feedback_2925dae3ab7d496ca438f506cae8baef&#34;&gt; &lt;span class=&#34;sr&#34;&gt;Some problems have options such as save, reset, hints, or show answer. These options follow the Submit button.&lt;/span&gt; &lt;/div&gt; &lt;/div&gt; &lt;div class=&#34;problem-action-buttons-wrapper&#34;&gt; &lt;/div&gt; &lt;/div&gt; &lt;div class=&#34;notification warning notification-gentle-alert is-hidden&#34; tabindex=&#34;-1&#34;&gt; &lt;span class=&#34;icon fa fa-exclamation-circle&#34; aria-hidden=&#34;true&#34;&gt;&lt;/span&gt; &lt;span class=&#34;notification-message&#34; aria-describedby=&#34;2925dae3ab7d496ca438f506cae8baef-problem-title&#34;&gt; &lt;/span&gt; &lt;div class=&#34;notification-btn-wrapper&#34;&gt; &lt;button type=&#34;button&#34; class=&#34;btn btn-default btn-small notification-btn review-btn sr&#34;&gt;Review&lt;/button&gt; &lt;/div&gt; &lt;/div&gt; &lt;div class=&#34;notification warning notification-save is-hidden&#34; tabindex=&#34;-1&#34;&gt; &lt;span class=&#34;icon fa fa-save&#34; aria-hidden=&#34;true&#34;&gt;&lt;/span&gt; &lt;span class=&#34;notification-message&#34; aria-describedby=&#34;2925dae3ab7d496ca438f506cae8baef-problem-title&#34;&gt;None &lt;/span&gt; &lt;div class=&#34;notification-btn-wrapper&#34;&gt; &lt;button type=&#34;button&#34; class=&#34;btn btn-default btn-small notification-btn review-btn sr&#34;&gt;Review&lt;/button&gt; &lt;/div&gt; &lt;/div&gt; &lt;div class=&#34;notification general notification-show-answer is-hidden&#34; tabindex=&#34;-1&#34;&gt; &lt;span class=&#34;icon fa fa-info-circle&#34; aria-hidden=&#34;true&#34;&gt;&lt;/span&gt; &lt;span class=&#34;notification-message&#34; aria-describedby=&#34;2925dae3ab7d496ca438f506cae8baef-problem-title&#34;&gt;Answers are displayed within the problem &lt;/span&gt; &lt;div class=&#34;notification-btn-wrapper&#34;&gt; &lt;button type=&#34;button&#34; class=&#34;btn btn-default btn-small notification-btn review-btn sr&#34;&gt;Review&lt;/button&gt; &lt;/div&gt; &lt;/div&gt; &lt;/div&gt; " data-graded="False"> <p class="loading-spinner"> <i class="fa fa-spinner fa-pulse fa-2x fa-fw"></i> <span class="sr">Loading&hellip;</span> </p> </div> </div> </div> <div class="vert vert-3" data-id="block-v1:MITx+15.053x+3T2016+type@problem+block@ffd6a5f5b999492f9fe35266cd27239b"> <div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-usage-id="block-v1:MITx+15.053x+3T2016+type@problem+block@ffd6a5f5b999492f9fe35266cd27239b" data-block-type="problem" data-graded="False" data-course-id="course-v1:MITx+15.053x+3T2016" data-has-score="True" data-request-token="6a146286852711ef8f57023e4242adeb" data-init="XBlockToXModuleShim" data-runtime-class="LmsRuntime" data-runtime-version="1"> <script type="json/xblock-args" class="xblock-json-init-args"> {"xmodule-type": "Problem"} </script> <div id="problem_ffd6a5f5b999492f9fe35266cd27239b" class="problems-wrapper" role="group" aria-labelledby="ffd6a5f5b999492f9fe35266cd27239b-problem-title" data-problem-id="block-v1:MITx+15.053x+3T2016+type@problem+block@ffd6a5f5b999492f9fe35266cd27239b" data-url="/courses/course-v1:MITx+15.053x+3T2016/xblock/block-v1:MITx+15.053x+3T2016+type@problem+block@ffd6a5f5b999492f9fe35266cd27239b/handler/xmodule_handler" data-problem-score="0" data-problem-total-possible="0" data-attempts-used="0" data-content=" &lt;h3 class=&#34;hd hd-3 problem-header&#34; id=&#34;ffd6a5f5b999492f9fe35266cd27239b-problem-title&#34; aria-describedby=&#34;block-v1:MITx+15.053x+3T2016+type@problem+block@ffd6a5f5b999492f9fe35266cd27239b-problem-progress&#34; tabindex=&#34;-1&#34;&gt; PART D &lt;/h3&gt; &lt;div class=&#34;problem-progress&#34; id=&#34;block-v1:MITx+15.053x+3T2016+type@problem+block@ffd6a5f5b999492f9fe35266cd27239b-problem-progress&#34;&gt;&lt;/div&gt; &lt;div class=&#34;problem&#34;&gt; &lt;div&gt; &lt;p&gt; &lt;a class=&#34;modal-content&#34; href=&#34;/assets/courseware/v1/6fb2f6bd7ed06236840b3b406ad2d78c/asset-v1:MITx+15.053x+3T2016+type@asset+block/R6P4dd.png&#34;&gt; &lt;img width=&#34;600&#34; height=&#34;439&#34; alt=&#34;Plot1&#34; src=&#34;/assets/courseware/v1/6fb2f6bd7ed06236840b3b406ad2d78c/asset-v1:MITx+15.053x+3T2016+type@asset+block/R6P4dd.png&#34;/&gt; &lt;/a&gt; &lt;/p&gt; &lt;p&gt; \(h(x) = \min\{|x-1| , x^2-3\} \) for \( -3 \le x \le 3. \) &lt;/p&gt; &lt;div class=&#34;wrapper-problem-response&#34; tabindex=&#34;-1&#34; aria-label=&#34;Question 1&#34; role=&#34;group&#34;&gt;&lt;div class=&#34;choicegroup capa_inputtype&#34; id=&#34;inputtype_ffd6a5f5b999492f9fe35266cd27239b_2_1&#34;&gt; &lt;fieldset aria-describedby=&#34;status_ffd6a5f5b999492f9fe35266cd27239b_2_1&#34;&gt; &lt;div class=&#34;field&#34;&gt; &lt;input type=&#34;radio&#34; name=&#34;input_ffd6a5f5b999492f9fe35266cd27239b_2_1&#34; id=&#34;input_ffd6a5f5b999492f9fe35266cd27239b_2_1_choice_0&#34; class=&#34;field-input input-radio&#34; value=&#34;choice_0&#34;/&gt;&lt;label id=&#34;ffd6a5f5b999492f9fe35266cd27239b_2_1-choice_0-label&#34; for=&#34;input_ffd6a5f5b999492f9fe35266cd27239b_2_1_choice_0&#34; class=&#34;response-label field-label label-inline&#34; aria-describedby=&#34;status_ffd6a5f5b999492f9fe35266cd27239b_2_1&#34;&gt; Convex &lt;/label&gt; &lt;/div&gt; &lt;div class=&#34;field&#34;&gt; &lt;input type=&#34;radio&#34; name=&#34;input_ffd6a5f5b999492f9fe35266cd27239b_2_1&#34; id=&#34;input_ffd6a5f5b999492f9fe35266cd27239b_2_1_choice_1&#34; class=&#34;field-input input-radio&#34; value=&#34;choice_1&#34;/&gt;&lt;label id=&#34;ffd6a5f5b999492f9fe35266cd27239b_2_1-choice_1-label&#34; for=&#34;input_ffd6a5f5b999492f9fe35266cd27239b_2_1_choice_1&#34; class=&#34;response-label field-label label-inline&#34; aria-describedby=&#34;status_ffd6a5f5b999492f9fe35266cd27239b_2_1&#34;&gt; Not Convex &lt;/label&gt; &lt;/div&gt; &lt;span id=&#34;answer_ffd6a5f5b999492f9fe35266cd27239b_2_1&#34;/&gt; &lt;/fieldset&gt; &lt;div class=&#34;indicator-container&#34;&gt; &lt;span class=&#34;status unanswered&#34; id=&#34;status_ffd6a5f5b999492f9fe35266cd27239b_2_1&#34; data-tooltip=&#34;Not yet answered.&#34;&gt; &lt;span class=&#34;sr&#34;&gt;unanswered&lt;/span&gt;&lt;span class=&#34;status-icon&#34; aria-hidden=&#34;true&#34;/&gt; &lt;/span&gt; &lt;/div&gt; &lt;/div&gt;&lt;/div&gt; &lt;div class=&#34;solution-span&#34;&gt; &lt;span id=&#34;solution_ffd6a5f5b999492f9fe35266cd27239b_solution_1&#34;/&gt; &lt;/div&gt;&lt;/div&gt; &lt;div class=&#34;action&#34;&gt; &lt;input type=&#34;hidden&#34; name=&#34;problem_id&#34; value=&#34;PART D&#34; /&gt; &lt;div class=&#34;submit-attempt-container&#34;&gt; &lt;button type=&#34;button&#34; class=&#34;submit btn-brand&#34; data-submitting=&#34;Submitting&#34; data-value=&#34;Submit&#34; data-should-enable-submit-button=&#34;True&#34; aria-describedby=&#34;submission_feedback_ffd6a5f5b999492f9fe35266cd27239b&#34; &gt; &lt;span class=&#34;submit-label&#34;&gt;Submit&lt;/span&gt; &lt;/button&gt; &lt;div class=&#34;submission-feedback&#34; id=&#34;submission_feedback_ffd6a5f5b999492f9fe35266cd27239b&#34;&gt; &lt;span class=&#34;sr&#34;&gt;Some problems have options such as save, reset, hints, or show answer. These options follow the Submit button.&lt;/span&gt; &lt;/div&gt; &lt;/div&gt; &lt;div class=&#34;problem-action-buttons-wrapper&#34;&gt; &lt;/div&gt; &lt;/div&gt; &lt;div class=&#34;notification warning notification-gentle-alert is-hidden&#34; tabindex=&#34;-1&#34;&gt; &lt;span class=&#34;icon fa fa-exclamation-circle&#34; aria-hidden=&#34;true&#34;&gt;&lt;/span&gt; &lt;span class=&#34;notification-message&#34; aria-describedby=&#34;ffd6a5f5b999492f9fe35266cd27239b-problem-title&#34;&gt; &lt;/span&gt; &lt;div class=&#34;notification-btn-wrapper&#34;&gt; &lt;button type=&#34;button&#34; class=&#34;btn btn-default btn-small notification-btn review-btn sr&#34;&gt;Review&lt;/button&gt; &lt;/div&gt; &lt;/div&gt; &lt;div class=&#34;notification warning notification-save is-hidden&#34; tabindex=&#34;-1&#34;&gt; &lt;span class=&#34;icon fa fa-save&#34; aria-hidden=&#34;true&#34;&gt;&lt;/span&gt; &lt;span class=&#34;notification-message&#34; aria-describedby=&#34;ffd6a5f5b999492f9fe35266cd27239b-problem-title&#34;&gt;None &lt;/span&gt; &lt;div class=&#34;notification-btn-wrapper&#34;&gt; &lt;button type=&#34;button&#34; class=&#34;btn btn-default btn-small notification-btn review-btn sr&#34;&gt;Review&lt;/button&gt; &lt;/div&gt; &lt;/div&gt; &lt;div class=&#34;notification general notification-show-answer is-hidden&#34; tabindex=&#34;-1&#34;&gt; &lt;span class=&#34;icon fa fa-info-circle&#34; aria-hidden=&#34;true&#34;&gt;&lt;/span&gt; &lt;span class=&#34;notification-message&#34; aria-describedby=&#34;ffd6a5f5b999492f9fe35266cd27239b-problem-title&#34;&gt;Answers are displayed within the problem &lt;/span&gt; &lt;div class=&#34;notification-btn-wrapper&#34;&gt; &lt;button type=&#34;button&#34; class=&#34;btn btn-default btn-small notification-btn review-btn sr&#34;&gt;Review&lt;/button&gt; &lt;/div&gt; &lt;/div&gt; &lt;/div&gt; " data-graded="False"> <p class="loading-spinner"> <i class="fa fa-spinner fa-pulse fa-2x fa-fw"></i> <span class="sr">Loading&hellip;</span> </p> </div> </div> </div> <div class="vert vert-4" data-id="block-v1:MITx+15.053x+3T2016+type@problem+block@2163a96241a14a63a5b4eebc0b224868"> <div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-usage-id="block-v1:MITx+15.053x+3T2016+type@problem+block@2163a96241a14a63a5b4eebc0b224868" data-block-type="problem" data-graded="False" data-course-id="course-v1:MITx+15.053x+3T2016" data-has-score="True" data-request-token="6a146286852711ef8f57023e4242adeb" data-init="XBlockToXModuleShim" data-runtime-class="LmsRuntime" data-runtime-version="1"> <script type="json/xblock-args" class="xblock-json-init-args"> {"xmodule-type": "Problem"} </script> <div id="problem_2163a96241a14a63a5b4eebc0b224868" class="problems-wrapper" role="group" aria-labelledby="2163a96241a14a63a5b4eebc0b224868-problem-title" data-problem-id="block-v1:MITx+15.053x+3T2016+type@problem+block@2163a96241a14a63a5b4eebc0b224868" data-url="/courses/course-v1:MITx+15.053x+3T2016/xblock/block-v1:MITx+15.053x+3T2016+type@problem+block@2163a96241a14a63a5b4eebc0b224868/handler/xmodule_handler" data-problem-score="0" data-problem-total-possible="0" data-attempts-used="0" data-content=" &lt;h3 class=&#34;hd hd-3 problem-header&#34; id=&#34;2163a96241a14a63a5b4eebc0b224868-problem-title&#34; aria-describedby=&#34;block-v1:MITx+15.053x+3T2016+type@problem+block@2163a96241a14a63a5b4eebc0b224868-problem-progress&#34; tabindex=&#34;-1&#34;&gt; PART E &lt;/h3&gt; &lt;div class=&#34;problem-progress&#34; id=&#34;block-v1:MITx+15.053x+3T2016+type@problem+block@2163a96241a14a63a5b4eebc0b224868-problem-progress&#34;&gt;&lt;/div&gt; &lt;div class=&#34;problem&#34;&gt; &lt;div&gt; &lt;p&gt; &lt;a class=&#34;modal-content&#34; href=&#34;/assets/courseware/v1/35933cdc22755a229e89bca9ef283591/asset-v1:MITx+15.053x+3T2016+type@asset+block/R6P4e2.png&#34;&gt; &lt;img width=&#34;600&#34; height=&#34;439&#34; alt=&#34;Plot1&#34; src=&#34;/assets/courseware/v1/35933cdc22755a229e89bca9ef283591/asset-v1:MITx+15.053x+3T2016+type@asset+block/R6P4e2.png&#34;/&gt; &lt;/a&gt; &lt;/p&gt; &lt;p&gt; \( h(x) = |x-1| - (x^2-3) \) for \( -3 \le x \le 3. \) &lt;/p&gt; &lt;div class=&#34;wrapper-problem-response&#34; tabindex=&#34;-1&#34; aria-label=&#34;Question 1&#34; role=&#34;group&#34;&gt;&lt;div class=&#34;choicegroup capa_inputtype&#34; id=&#34;inputtype_2163a96241a14a63a5b4eebc0b224868_2_1&#34;&gt; &lt;fieldset aria-describedby=&#34;status_2163a96241a14a63a5b4eebc0b224868_2_1&#34;&gt; &lt;div class=&#34;field&#34;&gt; &lt;input type=&#34;radio&#34; name=&#34;input_2163a96241a14a63a5b4eebc0b224868_2_1&#34; id=&#34;input_2163a96241a14a63a5b4eebc0b224868_2_1_choice_1&#34; class=&#34;field-input input-radio&#34; value=&#34;choice_1&#34;/&gt;&lt;label id=&#34;2163a96241a14a63a5b4eebc0b224868_2_1-choice_1-label&#34; for=&#34;input_2163a96241a14a63a5b4eebc0b224868_2_1_choice_1&#34; class=&#34;response-label field-label label-inline&#34; aria-describedby=&#34;status_2163a96241a14a63a5b4eebc0b224868_2_1&#34;&gt; Not Convex &lt;/label&gt; &lt;/div&gt; &lt;div class=&#34;field&#34;&gt; &lt;input type=&#34;radio&#34; name=&#34;input_2163a96241a14a63a5b4eebc0b224868_2_1&#34; id=&#34;input_2163a96241a14a63a5b4eebc0b224868_2_1_choice_0&#34; class=&#34;field-input input-radio&#34; value=&#34;choice_0&#34;/&gt;&lt;label id=&#34;2163a96241a14a63a5b4eebc0b224868_2_1-choice_0-label&#34; for=&#34;input_2163a96241a14a63a5b4eebc0b224868_2_1_choice_0&#34; class=&#34;response-label field-label label-inline&#34; aria-describedby=&#34;status_2163a96241a14a63a5b4eebc0b224868_2_1&#34;&gt; Convex &lt;/label&gt; &lt;/div&gt; &lt;span id=&#34;answer_2163a96241a14a63a5b4eebc0b224868_2_1&#34;/&gt; &lt;/fieldset&gt; &lt;div class=&#34;indicator-container&#34;&gt; &lt;span class=&#34;status unanswered&#34; id=&#34;status_2163a96241a14a63a5b4eebc0b224868_2_1&#34; data-tooltip=&#34;Not yet answered.&#34;&gt; &lt;span class=&#34;sr&#34;&gt;unanswered&lt;/span&gt;&lt;span class=&#34;status-icon&#34; aria-hidden=&#34;true&#34;/&gt; &lt;/span&gt; &lt;/div&gt; &lt;/div&gt;&lt;/div&gt; &lt;div class=&#34;solution-span&#34;&gt; &lt;span id=&#34;solution_2163a96241a14a63a5b4eebc0b224868_solution_1&#34;/&gt; &lt;/div&gt;&lt;/div&gt; &lt;div class=&#34;action&#34;&gt; &lt;input type=&#34;hidden&#34; name=&#34;problem_id&#34; value=&#34;PART E&#34; /&gt; &lt;div class=&#34;submit-attempt-container&#34;&gt; &lt;button type=&#34;button&#34; class=&#34;submit btn-brand&#34; data-submitting=&#34;Submitting&#34; data-value=&#34;Submit&#34; data-should-enable-submit-button=&#34;True&#34; aria-describedby=&#34;submission_feedback_2163a96241a14a63a5b4eebc0b224868&#34; &gt; &lt;span class=&#34;submit-label&#34;&gt;Submit&lt;/span&gt; &lt;/button&gt; &lt;div class=&#34;submission-feedback&#34; id=&#34;submission_feedback_2163a96241a14a63a5b4eebc0b224868&#34;&gt; &lt;span class=&#34;sr&#34;&gt;Some problems have options such as save, reset, hints, or show answer. These options follow the Submit button.&lt;/span&gt; &lt;/div&gt; &lt;/div&gt; &lt;div class=&#34;problem-action-buttons-wrapper&#34;&gt; &lt;/div&gt; &lt;/div&gt; &lt;div class=&#34;notification warning notification-gentle-alert is-hidden&#34; tabindex=&#34;-1&#34;&gt; &lt;span class=&#34;icon fa fa-exclamation-circle&#34; aria-hidden=&#34;true&#34;&gt;&lt;/span&gt; &lt;span class=&#34;notification-message&#34; aria-describedby=&#34;2163a96241a14a63a5b4eebc0b224868-problem-title&#34;&gt; &lt;/span&gt; &lt;div class=&#34;notification-btn-wrapper&#34;&gt; &lt;button type=&#34;button&#34; class=&#34;btn btn-default btn-small notification-btn review-btn sr&#34;&gt;Review&lt;/button&gt; &lt;/div&gt; &lt;/div&gt; &lt;div class=&#34;notification warning notification-save is-hidden&#34; tabindex=&#34;-1&#34;&gt; &lt;span class=&#34;icon fa fa-save&#34; aria-hidden=&#34;true&#34;&gt;&lt;/span&gt; &lt;span class=&#34;notification-message&#34; aria-describedby=&#34;2163a96241a14a63a5b4eebc0b224868-problem-title&#34;&gt;None &lt;/span&gt; &lt;div class=&#34;notification-btn-wrapper&#34;&gt; &lt;button type=&#34;button&#34; class=&#34;btn btn-default btn-small notification-btn review-btn sr&#34;&gt;Review&lt;/button&gt; &lt;/div&gt; &lt;/div&gt; &lt;div class=&#34;notification general notification-show-answer is-hidden&#34; tabindex=&#34;-1&#34;&gt; &lt;span class=&#34;icon fa fa-info-circle&#34; aria-hidden=&#34;true&#34;&gt;&lt;/span&gt; &lt;span class=&#34;notification-message&#34; aria-describedby=&#34;2163a96241a14a63a5b4eebc0b224868-problem-title&#34;&gt;Answers are displayed within the problem &lt;/span&gt; &lt;div class=&#34;notification-btn-wrapper&#34;&gt; &lt;button type=&#34;button&#34; class=&#34;btn btn-default btn-small notification-btn review-btn sr&#34;&gt;Review&lt;/button&gt; &lt;/div&gt; &lt;/div&gt; &lt;/div&gt; " data-graded="False"> <p class="loading-spinner"> <i class="fa fa-spinner fa-pulse fa-2x fa-fw"></i> <span class="sr">Loading&hellip;</span> </p> </div> </div> </div> </div> </div>