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<h2 class="hd hd-2 unit-title">Problem 1</h2>
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Part A
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<p>Consider the following linear program:</p>
<span>
<td class="equation">\[\left.\begin{array}{rrcl} \max &amp; x_1 + 2x_2 &amp; &amp; \\ \text {s.t.:} &amp; &amp; &amp; \\ &amp; x_1 &#8211; x_2 &amp; \ge &amp; -4 \\ &amp; x_1 + x_2 &amp; \le &amp; 8 \\ &amp; x_1 &amp; \le &amp; 5\\ &amp; x_2 &amp; \le &amp; 6 \\ &amp; x_1, x_2 &amp; \ge &amp; 0 \\ \end{array}\right\}\]</td>
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<p>Which of the following graphs shows the feasible region of the LP? The feasible region is indicated by the yellow area.</p>
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Part B
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<p>Are any of the above constraints redundant? If so, indicate which one(s). (For large linear programs, eliminating redundant constraints can sometimes speed up the solution of the linear program. Redundant constraints play an even more important role in integer programming.)</p>
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<input type="checkbox" name="input_cdc186483cc340fc97a466885f452c57_2_1[]" id="input_cdc186483cc340fc97a466885f452c57_2_1_choice_0" class="field-input input-checkbox" value="choice_0"/><label id="cdc186483cc340fc97a466885f452c57_2_1-choice_0-label" for="input_cdc186483cc340fc97a466885f452c57_2_1_choice_0" class="response-label field-label label-inline" aria-describedby="status_cdc186483cc340fc97a466885f452c57_2_1"> Constraint 1: [mathjaxinline] \ x_1 - x_2 \geq -4 [/mathjaxinline]
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<input type="checkbox" name="input_cdc186483cc340fc97a466885f452c57_2_1[]" id="input_cdc186483cc340fc97a466885f452c57_2_1_choice_1" class="field-input input-checkbox" value="choice_1"/><label id="cdc186483cc340fc97a466885f452c57_2_1-choice_1-label" for="input_cdc186483cc340fc97a466885f452c57_2_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_cdc186483cc340fc97a466885f452c57_2_1"> Constraint 2:[mathjaxinline] \ x_1 + x_2 \leq 8 [/mathjaxinline]
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<input type="checkbox" name="input_cdc186483cc340fc97a466885f452c57_2_1[]" id="input_cdc186483cc340fc97a466885f452c57_2_1_choice_2" class="field-input input-checkbox" value="choice_2"/><label id="cdc186483cc340fc97a466885f452c57_2_1-choice_2-label" for="input_cdc186483cc340fc97a466885f452c57_2_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_cdc186483cc340fc97a466885f452c57_2_1"> Constraint 3:[mathjaxinline] \ x_1 \leq 5 [/mathjaxinline]
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<input type="checkbox" name="input_cdc186483cc340fc97a466885f452c57_2_1[]" id="input_cdc186483cc340fc97a466885f452c57_2_1_choice_3" class="field-input input-checkbox" value="choice_3"/><label id="cdc186483cc340fc97a466885f452c57_2_1-choice_3-label" for="input_cdc186483cc340fc97a466885f452c57_2_1_choice_3" class="response-label field-label label-inline" aria-describedby="status_cdc186483cc340fc97a466885f452c57_2_1"> Constraint 4: [mathjaxinline] \ x_2 \leq 6 [/mathjaxinline]
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<input type="checkbox" name="input_cdc186483cc340fc97a466885f452c57_2_1[]" id="input_cdc186483cc340fc97a466885f452c57_2_1_choice_4" class="field-input input-checkbox" value="choice_4"/><label id="cdc186483cc340fc97a466885f452c57_2_1-choice_4-label" for="input_cdc186483cc340fc97a466885f452c57_2_1_choice_4" class="response-label field-label label-inline" aria-describedby="status_cdc186483cc340fc97a466885f452c57_2_1"> Constraint 5: [mathjaxinline] \ x_1, x_2 \geq 0 [/mathjaxinline]
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<input type="checkbox" name="input_cdc186483cc340fc97a466885f452c57_2_1[]" id="input_cdc186483cc340fc97a466885f452c57_2_1_choice_5" class="field-input input-checkbox" value="choice_5"/><label id="cdc186483cc340fc97a466885f452c57_2_1-choice_5-label" for="input_cdc186483cc340fc97a466885f452c57_2_1_choice_5" class="response-label field-label label-inline" aria-describedby="status_cdc186483cc340fc97a466885f452c57_2_1"> None of the above
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Part C
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<p>Solve the LP using the graphical method. Input the value \(x_1 \times x_2\), the product of the decision variables \(x_1, x_2\), at the optimal solution. </p>
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Part D
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<p>There is more than one optimal solution.</p>
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Part E
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<p> Suppose we add the constraint \(3x_1 + 2x_2 \geq \alpha\) to the LP. For which value(s) of \(\alpha\) is the constraint redundant? </p>
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Part F
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<p> Given the same constraint in Part E, for which value(s) of \( \alpha \) is the optimal solution found above no longer optimal?</p>
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Part G
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<p> Replace the objective function \( x_1 + 2x_2 \) with the objective function \(x_1 + \beta x_2\) What is the range of values of \(\beta\) for which the point \((2,6)\) is optimal?
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<h2 class="hd hd-2 unit-title">Problem 2</h2>
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PART A
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<p>Mr. Snowman, the mayor of Little Gotham, faces a serious snow storm in the coming days. He has two snow removal machines, and needs to remove snow from three different roads. He has 24 hours to remove 15 inches of snow from Road 1, 24 inches of snow from Road 2, and 18 inches of snow from Road 3. (Unusual for different amounts of snow to have to be removed from different roads, but that is the problem that Mr. Snowman faces.) In addition, each machine removes snow at a different rate on the three roads. </p>
<p>Machine 1 can remove one inch an hour from Road 1; it removes 1.2 inches an hour from Road 2; it removes two inches an hour from Road 3. Machine 2 can remove two inches an hour from Road 1, three inches an hour from Road 2 and three inches an hour from Road 3. The optimization problem is to remove as much snow as possible in 24 hours using the two machines.</p>
<p>The decision variables are:</p>
<ul>
<li>\(x_{rm}\): the number of hours that road r is cleaned by machine m.</li>
</ul>
<p>Mr. Snowman tries to model it as a linear program. He gives the following formulation.
As his assistant, you are concerned that some of the inequalities have been reversed. Which of the following is the best response?</p>
<span>
<p class="equation">\(\left. \begin{array}{rrcl} &amp; \max x_{11} + 2x_{12} + 1.2x_{21} + 3x_{22} + 2x_{31} +3x_{32} &amp; &amp; \\
\text {s.t.:} &amp; &amp; &amp; \\
(1)&amp; x_{11} + x_{21} + x_{31} &amp; \le &amp; 24 \\
(2)&amp; x_{12} + x_{22} + x_{32} &amp; \le &amp; 24 \\
(3)&amp; x_{11} + 2x_{12} &amp; \le &amp; 15 \\
(4)&amp; 1.2x_{21} + 3x_{22} &amp; \le &amp; 24 \\
(5)&amp; 2x_{31} + 3x_{32} &amp; \le &amp; 18 \\
(6)&amp; x_{11}, x_{12}, x_{21}, x_{22}, x_{31}, x_{32}, &amp; \ge &amp; 0 \\
\end{array}\right\}\)</p>
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The linear program is correct
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Linear inequalities (1) and (2) should be changed to \(\geq\)
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Linear inequalities (3), (4), and (5) should be changed to \(\geq\)
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Linear inequalities (1), (2), (3), (4), and (5) should all be changed to \(\geq\)
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PART B
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<p>Model the Linear Program above using Julia. (Recall that tutorials of how to use Julia and JuMP are provided in Recitation 0.) If you are having difficulty using Julia, you would get credit for modeling this problem using spreadsheet optimization. </p>
<p>How many inches of snow are removed by the 2 machines? Provide a rounded (not truncated) answer with 2 digits to the right of the decimal point.</p>
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PART C
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<p>Consider now that the mayor wants to remove the snow as soon as possible. </p>
<p>
Hint: Let \(z \) be the earliest time that machines \( m_1 \) and \( m_2 \) have both finished. The objective is to minimize \( z \).
</p>
<p>Mr. Snowman tries to model it as a linear program. He gives the following formulation.
As his assistant, you are concerned that some of the inequalities have been reversed. Which of the following is the best response?</p>
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<td class="equation">\(\left. \begin{array}{rrcl} &amp; \min z &amp; &amp; \\
\text {s.t.:} &amp; &amp; &amp; \\
(1)&amp; x_{11} + x_{21} + x_{31} &amp; \ge &amp; z\\
(2)&amp; x_{12} + x_{22} + x_{32} &amp; \ge &amp; z\\
(3)&amp; x_{11} + 2x_{12} &amp; = &amp; 15 \\
(4)&amp; 1.2x_{21} + 3x_{22} &amp; = &amp; 24 \\
(5)&amp; 2x_{31} + 3x_{32} &amp; = &amp; 18 \\
(6)&amp; x_{11}, x_{12}, x_{21}, x_{22}, x_{31}, x_{32}, z &amp; \ge &amp; 0 \\
\end{array}\right\}\)</td>
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The linear program is correct
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Linear inequalities (1) and (2) should be changed to \(\leq\)
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Linear inequalities (3), (4), and (5) should be changed to \(\geq\)
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Linear inequalities (3), (4), and (5) should be changed to \(\leq\)
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PART D
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<p>Model the Linear Program above using Julia. You may use spreadsheet optimization here if you used it in part B. </p>
<p>What is the minimum time needed to remove the snow? Use 2 digits to the right of the decimal point. (The answer should be rounded, not truncated.) </p>
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PART A
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<p>Consider a diet problem with two different foods (Food 1 and Food 2) and three different
nutrients (Nutrient 1, Nutrient 2, and Nutrient 3). The amount of ounces of each nutrient per
unit of food is given in Table 1. The price of Food 1 is $2. The price of Food 2 is $3. The
minimum requirements for the Nutrients 1, 2 and 3 are 30, 20, and 12 ounces, respectively.
The optimization problem is to find the minimum cost diet that meets the minimum nutritional
requirements.</p>
<center>
<table>
<tbody>
<tr>
<th scope="col" style="text-align: center; border: 1px solid black; padding: 10px;"/>
<th scope="col" style="text-align: center; border: 1px solid black; padding: 10px;">Food 1</th>
<th scope="col" style="text-align: center; border: 1px solid black; padding: 10px;">Food 2</th>
</tr>
<tr>
<th scope="row" style="text-align: center; border: 1px solid black; padding: 10px;">Nutrient 1</th>
<td style="text-align: center; border: 1px solid black; padding: 10px;">4</td>
<td style="text-align: center; border: 1px solid black; padding: 10px;">6</td>
</tr>
<tr>
<th scope="row" style="text-align: center; border: 1px solid black; padding: 10px;">Nutrient 2</th>
<td style="text-align: center; border: 1px solid black; padding: 10px;">6</td>
<td style="text-align: center; border: 1px solid black; padding: 10px;">2</td>
</tr>
<tr>
<th scope="row" style="text-align: center; border: 1px solid black; padding: 10px;">Nutrient 3</th>
<td style="text-align: center; border: 1px solid black; padding: 10px;">1</td>
<td style="text-align: center; border: 1px solid black; padding: 10px;">2</td>
</tr>
</tbody>
<caption>Table 1: Ounces of nutrients on each unit of food</caption>
</table>
</center>
<p/>
<p>The decision variables are</p>
<ul>
<li>x how much of food 1 to buy</li>
<li>y how much of food 2 to buy</li>
</ul>
<p>The formulation is</p>
<span>
<td class="equation">\(\left. \begin{array}{rrcl} &amp; \min 2x+3y &amp; &amp; \\
\text {s.t.:} &amp; &amp; &amp; \\
\text{Nutrient1} &amp; 4x+6y &amp; \ge &amp; 30 \\
\text{Nutrient2} &amp; 6x+2y &amp; \ge &amp; 20 \\
\text{Nutrient3} &amp; x+2y &amp; \ge &amp; 12 \\
\text{Non negativity} &amp; x,y &amp; \ge &amp; 0 \\
\end{array}\right\}\)</td>
</span>
<p>Solve the optimization problem in Julia. (You may use spreadsheet optimization instead. But we recommend, as before, that you try using Julia. At first, Julia is much more challenging to use than spreadsheets. In fact, this may continue to be true for a long time. The advantage of Julia is its flexibility, and its ability to deal with complex problems.)</p>
<p>Enter the sum of the optimal values of the decision variables. It will include 1 digit to the right of the decimal point (using rounding, not truncation). </p>
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PART C
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<p> Download <a class="modal-content" href="/assets/courseware/v1/e604d3fe385fa460022ad7ce23277545/asset-v1:MITx+15.053x+3T2016+type@asset+block/fooddata.csv"> fooddata.csv </a>, which contains randomly generated nutritional values for 10 different types of food.</p>
<p>We wish to minimize the number of calories consumed while intaking at least 50 fats, 300 carbohydrates, 60 proteins, and no more than 20 saturated fats. </p>
<p>Formulate a linear program, and determine the minimum number of calories consumed. The "read_csv" or "readtable" function in Julia will be helpful in reading the data set into memory.</p>
<p>Enter the (rounded) optimal number of calories (2 digits to the right of the decimal). </p>
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PART A
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Consider \(f(x): [ 0 ,5 ] \rightarrow \mathbb{R}\) a piecewise linear function such that the graph of the function corresponds to the segments between the points \(( 0 ,3 ),( 1 ,a ),( 3 ,1 ),( 4 ,2 ),( 5 ,5 ).\) \(a \in \mathbb{R} \text{ is a constant with a fixed value.} \)
<p>Below is a graph for the function under \(a =-2\),</p>
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<p>Is this function convex (assuming that \(a = -2 \) ) ?</p>
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<input type="radio" name="input_f06f7ff0f20a4ec58daf758e4a43ccb6_2_1" id="input_f06f7ff0f20a4ec58daf758e4a43ccb6_2_1_choice_1" class="field-input input-radio" value="choice_1"/><label id="f06f7ff0f20a4ec58daf758e4a43ccb6_2_1-choice_1-label" for="input_f06f7ff0f20a4ec58daf758e4a43ccb6_2_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_f06f7ff0f20a4ec58daf758e4a43ccb6_2_1"> Not convex
</label>
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<span id="answer_f06f7ff0f20a4ec58daf758e4a43ccb6_2_1"/>
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<h3 class="hd hd-3 problem-header" id="5c3c6506277f420583566d4233711cca-problem-title" aria-describedby="block-v1:MITx+15.053x+3T2016+type@problem+block@5c3c6506277f420583566d4233711cca-problem-progress" tabindex="-1">
PART B
</h3>
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<div>
<p>Below is a graph for the function under \(a =-1\),</p>
<span>
<p>
<a class="modal-content" href="/assets/courseware/v1/60b781f80039739d2df6fee0aeb2705d/asset-v1:MITx+15.053x+3T2016+type@asset+block/ps2p4_2.png">
<img width="500" height="500" alt="Plot1" src="/assets/courseware/v1/60b781f80039739d2df6fee0aeb2705d/asset-v1:MITx+15.053x+3T2016+type@asset+block/ps2p4_2.png"/>
</a>
</p>
</span>
<p>Is this function convex?</p>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><div class="choicegroup capa_inputtype" id="inputtype_5c3c6506277f420583566d4233711cca_2_1">
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<div class="field">
<input type="radio" name="input_5c3c6506277f420583566d4233711cca_2_1" id="input_5c3c6506277f420583566d4233711cca_2_1_choice_0" class="field-input input-radio" value="choice_0"/><label id="5c3c6506277f420583566d4233711cca_2_1-choice_0-label" for="input_5c3c6506277f420583566d4233711cca_2_1_choice_0" class="response-label field-label label-inline" aria-describedby="status_5c3c6506277f420583566d4233711cca_2_1"> Convex
</label>
</div>
<div class="field">
<input type="radio" name="input_5c3c6506277f420583566d4233711cca_2_1" id="input_5c3c6506277f420583566d4233711cca_2_1_choice_1" class="field-input input-radio" value="choice_1"/><label id="5c3c6506277f420583566d4233711cca_2_1-choice_1-label" for="input_5c3c6506277f420583566d4233711cca_2_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_5c3c6506277f420583566d4233711cca_2_1"> Not convex
</label>
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<span id="answer_5c3c6506277f420583566d4233711cca_2_1"/>
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<h3 class="hd hd-3 problem-header" id="dc662501c8404948bb128531490f9bb8-problem-title" aria-describedby="block-v1:MITx+15.053x+3T2016+type@problem+block@dc662501c8404948bb128531490f9bb8-problem-progress" tabindex="-1">
PART C
</h3>
<div class="problem-progress" id="block-v1:MITx+15.053x+3T2016+type@problem+block@dc662501c8404948bb128531490f9bb8-problem-progress"></div>
<div class="problem">
<div>
<p>Below is a graph for the function under \(a =\frac{7}{3}\),</p>
<span>
<p>
<a class="modal-content" href="/assets/courseware/v1/6f7634cf38cfcbccfe0b060907c8bf32/asset-v1:MITx+15.053x+3T2016+type@asset+block/ps2p4_3.png">
<img width="500" height="500" alt="Plot1" src="/assets/courseware/v1/6f7634cf38cfcbccfe0b060907c8bf32/asset-v1:MITx+15.053x+3T2016+type@asset+block/ps2p4_3.png"/>
</a>
</p>
</span>
<p>Is this function convex?</p>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><div class="choicegroup capa_inputtype" id="inputtype_dc662501c8404948bb128531490f9bb8_2_1">
<fieldset aria-describedby="status_dc662501c8404948bb128531490f9bb8_2_1">
<div class="field">
<input type="radio" name="input_dc662501c8404948bb128531490f9bb8_2_1" id="input_dc662501c8404948bb128531490f9bb8_2_1_choice_0" class="field-input input-radio" value="choice_0"/><label id="dc662501c8404948bb128531490f9bb8_2_1-choice_0-label" for="input_dc662501c8404948bb128531490f9bb8_2_1_choice_0" class="response-label field-label label-inline" aria-describedby="status_dc662501c8404948bb128531490f9bb8_2_1"> Convex
</label>
</div>
<div class="field">
<input type="radio" name="input_dc662501c8404948bb128531490f9bb8_2_1" id="input_dc662501c8404948bb128531490f9bb8_2_1_choice_1" class="field-input input-radio" value="choice_1"/><label id="dc662501c8404948bb128531490f9bb8_2_1-choice_1-label" for="input_dc662501c8404948bb128531490f9bb8_2_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_dc662501c8404948bb128531490f9bb8_2_1"> Not convex
</label>
</div>
<span id="answer_dc662501c8404948bb128531490f9bb8_2_1"/>
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data-problem-id="block-v1:MITx+15.053x+3T2016+type@problem+block@b7400f91ce83440bbaa5c06792169c73" data-url="/courses/course-v1:MITx+15.053x+3T2016/xblock/block-v1:MITx+15.053x+3T2016+type@problem+block@b7400f91ce83440bbaa5c06792169c73/handler/xmodule_handler"
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<h3 class="hd hd-3 problem-header" id="b7400f91ce83440bbaa5c06792169c73-problem-title" aria-describedby="block-v1:MITx+15.053x+3T2016+type@problem+block@b7400f91ce83440bbaa5c06792169c73-problem-progress" tabindex="-1">
PART D
</h3>
<div class="problem-progress" id="block-v1:MITx+15.053x+3T2016+type@problem+block@b7400f91ce83440bbaa5c06792169c73-problem-progress"></div>
<div class="problem">
<div>
<p>Below is a graph for the function under \(a =4\),</p>
<span>
<p>
<a class="modal-content" href="/assets/courseware/v1/608a0f313189b64c4d87400186faaa98/asset-v1:MITx+15.053x+3T2016+type@asset+block/ps2p4_4.png">
<img width="500" height="500" alt="Plot1" src="/assets/courseware/v1/608a0f313189b64c4d87400186faaa98/asset-v1:MITx+15.053x+3T2016+type@asset+block/ps2p4_4.png"/>
</a>
</p>
</span>
<p>Is this function convex?</p>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><div class="choicegroup capa_inputtype" id="inputtype_b7400f91ce83440bbaa5c06792169c73_2_1">
<fieldset aria-describedby="status_b7400f91ce83440bbaa5c06792169c73_2_1">
<div class="field">
<input type="radio" name="input_b7400f91ce83440bbaa5c06792169c73_2_1" id="input_b7400f91ce83440bbaa5c06792169c73_2_1_choice_0" class="field-input input-radio" value="choice_0"/><label id="b7400f91ce83440bbaa5c06792169c73_2_1-choice_0-label" for="input_b7400f91ce83440bbaa5c06792169c73_2_1_choice_0" class="response-label field-label label-inline" aria-describedby="status_b7400f91ce83440bbaa5c06792169c73_2_1"> Convex
</label>
</div>
<div class="field">
<input type="radio" name="input_b7400f91ce83440bbaa5c06792169c73_2_1" id="input_b7400f91ce83440bbaa5c06792169c73_2_1_choice_1" class="field-input input-radio" value="choice_1"/><label id="b7400f91ce83440bbaa5c06792169c73_2_1-choice_1-label" for="input_b7400f91ce83440bbaa5c06792169c73_2_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_b7400f91ce83440bbaa5c06792169c73_2_1"> Not convex
</label>
</div>
<span id="answer_b7400f91ce83440bbaa5c06792169c73_2_1"/>
</fieldset>
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<span class="status unanswered" id="status_b7400f91ce83440bbaa5c06792169c73_2_1" data-tooltip="Not yet answered.">
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PART E
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<p> Based on the previous question, what is the range of \(a\) for \(f(x)\) to be a convex function?</p>
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PART F
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<p>We are interested in solving the optimization problem: \( min_{x \in [0,5]} f(x)\) in the case that the function is convex. (This technique does not necessarily work if the function is not convex.) To do this, we first break up \(f\) into four segments and and find the linear functions corresponding to each segment. We denote the four segments as \(f_1 (x),f_2 (x),f_3 (x),f_4 (x)\) respectively from left to right. </p>
<p>Which of the following describe the segment going through points \((0,1)\) to \((1,a)\)?</p>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><div class="choicegroup capa_inputtype" id="inputtype_01e222861c1649c79a0ed385b7f4f5af_2_1">
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<input type="radio" name="input_01e222861c1649c79a0ed385b7f4f5af_2_1" id="input_01e222861c1649c79a0ed385b7f4f5af_2_1_choice_0" class="field-input input-radio" value="choice_0"/><label id="01e222861c1649c79a0ed385b7f4f5af_2_1-choice_0-label" for="input_01e222861c1649c79a0ed385b7f4f5af_2_1_choice_0" class="response-label field-label label-inline" aria-describedby="status_01e222861c1649c79a0ed385b7f4f5af_2_1"> \(f_1(x) = (a+1)x + 1\)
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<input type="radio" name="input_01e222861c1649c79a0ed385b7f4f5af_2_1" id="input_01e222861c1649c79a0ed385b7f4f5af_2_1_choice_1" class="field-input input-radio" value="choice_1"/><label id="01e222861c1649c79a0ed385b7f4f5af_2_1-choice_1-label" for="input_01e222861c1649c79a0ed385b7f4f5af_2_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_01e222861c1649c79a0ed385b7f4f5af_2_1"> \(f_1(x) = (a-1)x + 1\)
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<input type="radio" name="input_01e222861c1649c79a0ed385b7f4f5af_2_1" id="input_01e222861c1649c79a0ed385b7f4f5af_2_1_choice_2" class="field-input input-radio" value="choice_2"/><label id="01e222861c1649c79a0ed385b7f4f5af_2_1-choice_2-label" for="input_01e222861c1649c79a0ed385b7f4f5af_2_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_01e222861c1649c79a0ed385b7f4f5af_2_1"> \(f_1(x) = (a-1)x - 1\)
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<input type="radio" name="input_01e222861c1649c79a0ed385b7f4f5af_2_1" id="input_01e222861c1649c79a0ed385b7f4f5af_2_1_choice_3" class="field-input input-radio" value="choice_3"/><label id="01e222861c1649c79a0ed385b7f4f5af_2_1-choice_3-label" for="input_01e222861c1649c79a0ed385b7f4f5af_2_1_choice_3" class="response-label field-label label-inline" aria-describedby="status_01e222861c1649c79a0ed385b7f4f5af_2_1"> \(f_1(x) = (a+1)x - 1\)
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PART G
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<p>Which of the following describe the segment going through points \((1,a)\) to \((3,1)\)?</p>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><div class="choicegroup capa_inputtype" id="inputtype_aebc5fc103684ecbbeb160b7a235a352_2_1">
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<input type="radio" name="input_aebc5fc103684ecbbeb160b7a235a352_2_1" id="input_aebc5fc103684ecbbeb160b7a235a352_2_1_choice_0" class="field-input input-radio" value="choice_0"/><label id="aebc5fc103684ecbbeb160b7a235a352_2_1-choice_0-label" for="input_aebc5fc103684ecbbeb160b7a235a352_2_1_choice_0" class="response-label field-label label-inline" aria-describedby="status_aebc5fc103684ecbbeb160b7a235a352_2_1"> \(f_2(x) = \frac{1+a}{2}x + \frac{3a-1}{2}\)
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<input type="radio" name="input_aebc5fc103684ecbbeb160b7a235a352_2_1" id="input_aebc5fc103684ecbbeb160b7a235a352_2_1_choice_1" class="field-input input-radio" value="choice_1"/><label id="aebc5fc103684ecbbeb160b7a235a352_2_1-choice_1-label" for="input_aebc5fc103684ecbbeb160b7a235a352_2_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_aebc5fc103684ecbbeb160b7a235a352_2_1"> \(f_2(x) = \frac{1-a}{2}x + \frac{3a+1}{2}\)
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<input type="radio" name="input_aebc5fc103684ecbbeb160b7a235a352_2_1" id="input_aebc5fc103684ecbbeb160b7a235a352_2_1_choice_2" class="field-input input-radio" value="choice_2"/><label id="aebc5fc103684ecbbeb160b7a235a352_2_1-choice_2-label" for="input_aebc5fc103684ecbbeb160b7a235a352_2_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_aebc5fc103684ecbbeb160b7a235a352_2_1"> \(f_2(x) = \frac{1+a}{2}x + \frac{3a+1}{2}\)
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<input type="radio" name="input_aebc5fc103684ecbbeb160b7a235a352_2_1" id="input_aebc5fc103684ecbbeb160b7a235a352_2_1_choice_3" class="field-input input-radio" value="choice_3"/><label id="aebc5fc103684ecbbeb160b7a235a352_2_1-choice_3-label" for="input_aebc5fc103684ecbbeb160b7a235a352_2_1_choice_3" class="response-label field-label label-inline" aria-describedby="status_aebc5fc103684ecbbeb160b7a235a352_2_1"> \(f_2(x) = \frac{1-a}{2}x + \frac{3a-1}{2}\)
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PART H
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<p>Which of the following describe the segment going through points \((3,1)\) to \((4,2)\)?</p>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><div class="choicegroup capa_inputtype" id="inputtype_bde1fb496fad412d8a4a5175b1ed590d_2_1">
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<input type="radio" name="input_bde1fb496fad412d8a4a5175b1ed590d_2_1" id="input_bde1fb496fad412d8a4a5175b1ed590d_2_1_choice_0" class="field-input input-radio" value="choice_0"/><label id="bde1fb496fad412d8a4a5175b1ed590d_2_1-choice_0-label" for="input_bde1fb496fad412d8a4a5175b1ed590d_2_1_choice_0" class="response-label field-label label-inline" aria-describedby="status_bde1fb496fad412d8a4a5175b1ed590d_2_1"> \(f_3(x) = x-2\)
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<input type="radio" name="input_bde1fb496fad412d8a4a5175b1ed590d_2_1" id="input_bde1fb496fad412d8a4a5175b1ed590d_2_1_choice_1" class="field-input input-radio" value="choice_1"/><label id="bde1fb496fad412d8a4a5175b1ed590d_2_1-choice_1-label" for="input_bde1fb496fad412d8a4a5175b1ed590d_2_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_bde1fb496fad412d8a4a5175b1ed590d_2_1"> \(f_3(x) = x+2\)
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<input type="radio" name="input_bde1fb496fad412d8a4a5175b1ed590d_2_1" id="input_bde1fb496fad412d8a4a5175b1ed590d_2_1_choice_2" class="field-input input-radio" value="choice_2"/><label id="bde1fb496fad412d8a4a5175b1ed590d_2_1-choice_2-label" for="input_bde1fb496fad412d8a4a5175b1ed590d_2_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_bde1fb496fad412d8a4a5175b1ed590d_2_1"> \(f_3(x) = -x+2\)
</label>
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<input type="radio" name="input_bde1fb496fad412d8a4a5175b1ed590d_2_1" id="input_bde1fb496fad412d8a4a5175b1ed590d_2_1_choice_3" class="field-input input-radio" value="choice_3"/><label id="bde1fb496fad412d8a4a5175b1ed590d_2_1-choice_3-label" for="input_bde1fb496fad412d8a4a5175b1ed590d_2_1_choice_3" class="response-label field-label label-inline" aria-describedby="status_bde1fb496fad412d8a4a5175b1ed590d_2_1"> \(f_3(x) = -x-2\)
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PART I
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<p>Which of the following describe the segment going through points \((4,2)\) to \((5,5)\)?</p>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><div class="choicegroup capa_inputtype" id="inputtype_47a8f5ba7d364d479516ed3270bbe8fc_2_1">
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<input type="radio" name="input_47a8f5ba7d364d479516ed3270bbe8fc_2_1" id="input_47a8f5ba7d364d479516ed3270bbe8fc_2_1_choice_0" class="field-input input-radio" value="choice_0"/><label id="47a8f5ba7d364d479516ed3270bbe8fc_2_1-choice_0-label" for="input_47a8f5ba7d364d479516ed3270bbe8fc_2_1_choice_0" class="response-label field-label label-inline" aria-describedby="status_47a8f5ba7d364d479516ed3270bbe8fc_2_1"> \(f_4(x) = 3x+10\)
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<input type="radio" name="input_47a8f5ba7d364d479516ed3270bbe8fc_2_1" id="input_47a8f5ba7d364d479516ed3270bbe8fc_2_1_choice_1" class="field-input input-radio" value="choice_1"/><label id="47a8f5ba7d364d479516ed3270bbe8fc_2_1-choice_1-label" for="input_47a8f5ba7d364d479516ed3270bbe8fc_2_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_47a8f5ba7d364d479516ed3270bbe8fc_2_1"> \(f_4(x) = -3x+10\)
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<input type="radio" name="input_47a8f5ba7d364d479516ed3270bbe8fc_2_1" id="input_47a8f5ba7d364d479516ed3270bbe8fc_2_1_choice_2" class="field-input input-radio" value="choice_2"/><label id="47a8f5ba7d364d479516ed3270bbe8fc_2_1-choice_2-label" for="input_47a8f5ba7d364d479516ed3270bbe8fc_2_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_47a8f5ba7d364d479516ed3270bbe8fc_2_1"> \(f_4(x) = 3x-10\)
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<input type="radio" name="input_47a8f5ba7d364d479516ed3270bbe8fc_2_1" id="input_47a8f5ba7d364d479516ed3270bbe8fc_2_1_choice_3" class="field-input input-radio" value="choice_3"/><label id="47a8f5ba7d364d479516ed3270bbe8fc_2_1-choice_3-label" for="input_47a8f5ba7d364d479516ed3270bbe8fc_2_1_choice_3" class="response-label field-label label-inline" aria-describedby="status_47a8f5ba7d364d479516ed3270bbe8fc_2_1"> \(f_4(x) = -3x-10\)
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PART J
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<p>Suppose a is under the range which make \(f(x)\) convex. We can then turn \(min_{x \in [0,5]} \ f(x)\) into the following min-max problem</p>
<p>min max \( \{f_1(x), f_2(x),f_3(x), f_4(x) \}\) </p>
<span>
<td class="equation">\[\left.\begin{array}{rrcl} \text{min max} &amp; \{(a-1)x+1, \frac{1-a}{2}x + \frac{3a-1}{2}, x-2, 3x-10\} \\
\text {s.t.:} &amp; &amp; &amp; \\
&amp; x \leq 5 \\
&amp; x \geq 0 \end{array}\right\}\]</td>
</span>
<p>Make this min-max problem into a linear programming model</p>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><div class="choicegroup capa_inputtype" id="inputtype_3523ff849d334a4490fd50fb20aa2349_2_1">
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<input type="radio" name="input_3523ff849d334a4490fd50fb20aa2349_2_1" id="input_3523ff849d334a4490fd50fb20aa2349_2_1_choice_0" class="field-input input-radio" value="choice_0"/><label id="3523ff849d334a4490fd50fb20aa2349_2_1-choice_0-label" for="input_3523ff849d334a4490fd50fb20aa2349_2_1_choice_0" class="response-label field-label label-inline" aria-describedby="status_3523ff849d334a4490fd50fb20aa2349_2_1">
<text>
<td class="equation">\[\left.\begin{array}{rrcl} \min &amp; y \\
\text {s.t.:} &amp; &amp; &amp; \\
&amp; x \leq 5 \\
&amp; y \geq (a-1)x + 1 \\
&amp; y \geq \frac{1-a}{2}x + \frac{3a-1}{2} \\
&amp; y \leq x-2 \\
&amp; y \geq 3x-10 \\
&amp; x \geq 0 \end{array}\right\}\]</td>
</text>
</label>
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<input type="radio" name="input_3523ff849d334a4490fd50fb20aa2349_2_1" id="input_3523ff849d334a4490fd50fb20aa2349_2_1_choice_1" class="field-input input-radio" value="choice_1"/><label id="3523ff849d334a4490fd50fb20aa2349_2_1-choice_1-label" for="input_3523ff849d334a4490fd50fb20aa2349_2_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_3523ff849d334a4490fd50fb20aa2349_2_1">
<text>
<td class="equation">\[\left.\begin{array}{rrcl} \min &amp; y \\
\text {s.t.:} &amp; &amp; &amp; \\
&amp; y \geq (a-1)x+1 \\
&amp; y \geq \frac{1-a}{2}x + \frac{3a-1}{2} \\
&amp; y \geq x-2 \\
&amp; y \geq 3x-10 \\
&amp; x \geq 0 \end{array}\right\}\]</td>
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<input type="radio" name="input_3523ff849d334a4490fd50fb20aa2349_2_1" id="input_3523ff849d334a4490fd50fb20aa2349_2_1_choice_2" class="field-input input-radio" value="choice_2"/><label id="3523ff849d334a4490fd50fb20aa2349_2_1-choice_2-label" for="input_3523ff849d334a4490fd50fb20aa2349_2_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_3523ff849d334a4490fd50fb20aa2349_2_1">
<text>
<td class="equation">\[\left.\begin{array}{rrcl} \min &amp; y \\
\text {s.t.:} &amp; &amp; &amp; \\
&amp; x \leq 5 \\
&amp; y \geq (a-1)x+1 \\
&amp; y \geq \frac{1-a}{2}x + \frac{3a-1}{2} \\
&amp; y \geq x-2 \\
&amp; y \geq 3x-10 \\
&amp; x \geq 0 \end{array}\right\}\]</td>
</text>
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<input type="radio" name="input_3523ff849d334a4490fd50fb20aa2349_2_1" id="input_3523ff849d334a4490fd50fb20aa2349_2_1_choice_3" class="field-input input-radio" value="choice_3"/><label id="3523ff849d334a4490fd50fb20aa2349_2_1-choice_3-label" for="input_3523ff849d334a4490fd50fb20aa2349_2_1_choice_3" class="response-label field-label label-inline" aria-describedby="status_3523ff849d334a4490fd50fb20aa2349_2_1">
<text>
<td class="equation">\[\left.\begin{array}{rrcl} \min &amp; y \\
\text {s.t.:} &amp; &amp; &amp; \\
&amp; x \leq 5 \\
&amp; y \geq (a-1)x+1 \\
&amp; y \geq \frac{1-a}{2}x - \frac{3a-1}{2} \\
&amp; y \geq x+2 \\
&amp; y \geq 3x-10 \\
&amp; x \geq 0 \end{array}\right\}\]</td>
</text>
</label>
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<h2 class="hd hd-2 unit-title">Problem 5</h2>
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PART A
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<div><p>For each function and accompanying graph, determine whether the function is convex or not.</p><p><a class="modal-content" href="/assets/courseware/v1/38bcd85ebb5a454fa9a0d962a2001db6/asset-v1:MITx+15.053x+3T2016+type@asset+block/convex1.png"><img width="600" height="439" alt="Plot1" src="/assets/courseware/v1/38bcd85ebb5a454fa9a0d962a2001db6/asset-v1:MITx+15.053x+3T2016+type@asset+block/convex1.png"/></a></p>
\(f(x) =2x-4\)
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Not Convex
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Convex
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PART B
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<div><p><a class="modal-content" href="/assets/courseware/v1/dbcdaec9a3d8d3869618a51f695f8f09/asset-v1:MITx+15.053x+3T2016+type@asset+block/convex2.png"><img width="600" height="439" alt="Plot1" src="/assets/courseware/v1/dbcdaec9a3d8d3869618a51f695f8f09/asset-v1:MITx+15.053x+3T2016+type@asset+block/convex2.png"/></a></p>
\(f(x) = \vert x-2 \vert\)
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Not Convex
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Convex
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PART C
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<div><p><a class="modal-content" href="/assets/courseware/v1/aa7c3278d08a60a661317c181684f411/asset-v1:MITx+15.053x+3T2016+type@asset+block/convex3.png"><img width="600" height="439" alt="Plot1" src="/assets/courseware/v1/aa7c3278d08a60a661317c181684f411/asset-v1:MITx+15.053x+3T2016+type@asset+block/convex3.png"/></a></p>
\(f(x) = -\vert x \vert\)
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Not Convex
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Convex
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PART D
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<div><p><a class="modal-content" href="/assets/courseware/v1/3e61018ae1e88da0a841d67a69c56236/asset-v1:MITx+15.053x+3T2016+type@asset+block/convex4.png"><img width="600" height="439" alt="Plot1" src="/assets/courseware/v1/3e61018ae1e88da0a841d67a69c56236/asset-v1:MITx+15.053x+3T2016+type@asset+block/convex4.png"/></a></p>
\(f(x) = x^2 - x +3\)
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Not Convex
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Convex
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PART E
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\(f(x) = x^3 - 3x^2 +4x - 100\)
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<input type="radio" name="input_9e646e9980b84cc1ab26b13195ffc5bb_2_1" id="input_9e646e9980b84cc1ab26b13195ffc5bb_2_1_choice_1a#B" class="field-input input-radio" value="choice_1a#B"/><label id="9e646e9980b84cc1ab26b13195ffc5bb_2_1-choice_1a#B-label" for="input_9e646e9980b84cc1ab26b13195ffc5bb_2_1_choice_1a#B" class="response-label field-label label-inline" aria-describedby="status_9e646e9980b84cc1ab26b13195ffc5bb_2_1">
Not Convex
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<input type="radio" name="input_9e646e9980b84cc1ab26b13195ffc5bb_2_1" id="input_9e646e9980b84cc1ab26b13195ffc5bb_2_1_choice_1a#A" class="field-input input-radio" value="choice_1a#A"/><label id="9e646e9980b84cc1ab26b13195ffc5bb_2_1-choice_1a#A-label" for="input_9e646e9980b84cc1ab26b13195ffc5bb_2_1_choice_1a#A" class="response-label field-label label-inline" aria-describedby="status_9e646e9980b84cc1ab26b13195ffc5bb_2_1">
Convex
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PART F
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<div><p><a class="modal-content" href="/assets/courseware/v1/6aeb665b2d106ff6d13d14b2a4d55f08/asset-v1:MITx+15.053x+3T2016+type@asset+block/convex6.png"><img width="600" height="439" alt="Plot1" src="/assets/courseware/v1/6aeb665b2d106ff6d13d14b2a4d55f08/asset-v1:MITx+15.053x+3T2016+type@asset+block/convex6.png"/></a></p>
\(f(x) = e^{x}\)
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Not Convex
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<input type="radio" name="input_357d3a67248b4f0eb02b5b3a99075a4c_2_1" id="input_357d3a67248b4f0eb02b5b3a99075a4c_2_1_choice_1a#A" class="field-input input-radio" value="choice_1a#A"/><label id="357d3a67248b4f0eb02b5b3a99075a4c_2_1-choice_1a#A-label" for="input_357d3a67248b4f0eb02b5b3a99075a4c_2_1_choice_1a#A" class="response-label field-label label-inline" aria-describedby="status_357d3a67248b4f0eb02b5b3a99075a4c_2_1">
Convex
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PART G
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\(f(x) = \frac{1}{x}\)
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Convex
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PART H
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\(f(x) = \frac{1}{x} \text{ for } x \gt 0\)
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><div class="choicegroup capa_inputtype" id="inputtype_7317865aa92842d99246897b059c6e24_2_1">
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Not Convex
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Convex
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PART I
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\(f(x) = max \{x-3, -3x+5\}\)
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Not Convex
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Convex
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PART J
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\(f(x) = min \{x-3, -3x+5\}\)
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Not Convex
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PART K
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\(f(x) = \lfloor x \rfloor, \text{ which is the largest integer not greater than x } \)
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Convex
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PART L
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\(f(x) = sin(x)\)
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Not Convex
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Convex
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PART M
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\( f(x) = sin(x), \pi \le x \le 2 \pi\)
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Not Convex
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Convex
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<h2 class="hd hd-2 unit-title">Problem 6</h2>
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<h3 class="hd hd-3 problem-header" id="f9fbccf94fa54278a94df4de2b19c08a-problem-title" aria-describedby="block-v1:MITx+15.053x+3T2016+type@problem+block@f9fbccf94fa54278a94df4de2b19c08a-problem-progress" tabindex="-1">
PART A
</h3>
<div class="problem-progress" id="block-v1:MITx+15.053x+3T2016+type@problem+block@f9fbccf94fa54278a94df4de2b19c08a-problem-progress"></div>
<div class="problem">
<div><p>The following feasible region described is convex:</p>
\(\{(x,y): x+y \lt 1, x \gt 0, y \gt 0 \}\)
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><div class="choicegroup capa_inputtype" id="inputtype_f9fbccf94fa54278a94df4de2b19c08a_2_1">
<fieldset aria-describedby="status_f9fbccf94fa54278a94df4de2b19c08a_2_1">
<div class="field">
<input type="radio" name="input_f9fbccf94fa54278a94df4de2b19c08a_2_1" id="input_f9fbccf94fa54278a94df4de2b19c08a_2_1_choice_1a#A" class="field-input input-radio" value="choice_1a#A"/><label id="f9fbccf94fa54278a94df4de2b19c08a_2_1-choice_1a#A-label" for="input_f9fbccf94fa54278a94df4de2b19c08a_2_1_choice_1a#A" class="response-label field-label label-inline" aria-describedby="status_f9fbccf94fa54278a94df4de2b19c08a_2_1">
True
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</div>
<div class="field">
<input type="radio" name="input_f9fbccf94fa54278a94df4de2b19c08a_2_1" id="input_f9fbccf94fa54278a94df4de2b19c08a_2_1_choice_1a#B" class="field-input input-radio" value="choice_1a#B"/><label id="f9fbccf94fa54278a94df4de2b19c08a_2_1-choice_1a#B-label" for="input_f9fbccf94fa54278a94df4de2b19c08a_2_1_choice_1a#B" class="response-label field-label label-inline" aria-describedby="status_f9fbccf94fa54278a94df4de2b19c08a_2_1">
False
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<span class="status unanswered" id="status_f9fbccf94fa54278a94df4de2b19c08a_2_1" data-tooltip="Not yet answered.">
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<span id="solution_f9fbccf94fa54278a94df4de2b19c08a_solution_1"/>
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<h3 class="hd hd-3 problem-header" id="815d0aa0bbde4361a1bd0364cfd3b2d1-problem-title" aria-describedby="block-v1:MITx+15.053x+3T2016+type@problem+block@815d0aa0bbde4361a1bd0364cfd3b2d1-problem-progress" tabindex="-1">
PART B
</h3>
<div class="problem-progress" id="block-v1:MITx+15.053x+3T2016+type@problem+block@815d0aa0bbde4361a1bd0364cfd3b2d1-problem-progress"></div>
<div class="problem">
<div><p>The following feasible region described is convex:</p>
\(\{(x,y): x+y = 1, x \gt 0, y \gt 0 \}\)
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><div class="choicegroup capa_inputtype" id="inputtype_815d0aa0bbde4361a1bd0364cfd3b2d1_2_1">
<fieldset aria-describedby="status_815d0aa0bbde4361a1bd0364cfd3b2d1_2_1">
<div class="field">
<input type="radio" name="input_815d0aa0bbde4361a1bd0364cfd3b2d1_2_1" id="input_815d0aa0bbde4361a1bd0364cfd3b2d1_2_1_choice_1a#A" class="field-input input-radio" value="choice_1a#A"/><label id="815d0aa0bbde4361a1bd0364cfd3b2d1_2_1-choice_1a#A-label" for="input_815d0aa0bbde4361a1bd0364cfd3b2d1_2_1_choice_1a#A" class="response-label field-label label-inline" aria-describedby="status_815d0aa0bbde4361a1bd0364cfd3b2d1_2_1">
True
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</div>
<div class="field">
<input type="radio" name="input_815d0aa0bbde4361a1bd0364cfd3b2d1_2_1" id="input_815d0aa0bbde4361a1bd0364cfd3b2d1_2_1_choice_1a#B" class="field-input input-radio" value="choice_1a#B"/><label id="815d0aa0bbde4361a1bd0364cfd3b2d1_2_1-choice_1a#B-label" for="input_815d0aa0bbde4361a1bd0364cfd3b2d1_2_1_choice_1a#B" class="response-label field-label label-inline" aria-describedby="status_815d0aa0bbde4361a1bd0364cfd3b2d1_2_1">
False
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</div>
<span id="answer_815d0aa0bbde4361a1bd0364cfd3b2d1_2_1"/>
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<span class="status unanswered" id="status_815d0aa0bbde4361a1bd0364cfd3b2d1_2_1" data-tooltip="Not yet answered.">
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</div>
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<span id="solution_815d0aa0bbde4361a1bd0364cfd3b2d1_solution_1"/>
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<input type="hidden" name="problem_id" value="PART B" />
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data-problem-id="block-v1:MITx+15.053x+3T2016+type@problem+block@356fa8a9fa894a6db8a9717b0ea62d5e" data-url="/courses/course-v1:MITx+15.053x+3T2016/xblock/block-v1:MITx+15.053x+3T2016+type@problem+block@356fa8a9fa894a6db8a9717b0ea62d5e/handler/xmodule_handler"
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PART C
</h3>
<div class="problem-progress" id="block-v1:MITx+15.053x+3T2016+type@problem+block@356fa8a9fa894a6db8a9717b0ea62d5e-problem-progress"></div>
<div class="problem">
<div><p>The following feasible region described is convex:</p>
\(\{(x,y): \vert x \vert + \vert y \vert = 1 \}\)
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><div class="choicegroup capa_inputtype" id="inputtype_356fa8a9fa894a6db8a9717b0ea62d5e_2_1">
<fieldset aria-describedby="status_356fa8a9fa894a6db8a9717b0ea62d5e_2_1">
<div class="field">
<input type="radio" name="input_356fa8a9fa894a6db8a9717b0ea62d5e_2_1" id="input_356fa8a9fa894a6db8a9717b0ea62d5e_2_1_choice_1a#A" class="field-input input-radio" value="choice_1a#A"/><label id="356fa8a9fa894a6db8a9717b0ea62d5e_2_1-choice_1a#A-label" for="input_356fa8a9fa894a6db8a9717b0ea62d5e_2_1_choice_1a#A" class="response-label field-label label-inline" aria-describedby="status_356fa8a9fa894a6db8a9717b0ea62d5e_2_1">
True
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</div>
<div class="field">
<input type="radio" name="input_356fa8a9fa894a6db8a9717b0ea62d5e_2_1" id="input_356fa8a9fa894a6db8a9717b0ea62d5e_2_1_choice_1a#B" class="field-input input-radio" value="choice_1a#B"/><label id="356fa8a9fa894a6db8a9717b0ea62d5e_2_1-choice_1a#B-label" for="input_356fa8a9fa894a6db8a9717b0ea62d5e_2_1_choice_1a#B" class="response-label field-label label-inline" aria-describedby="status_356fa8a9fa894a6db8a9717b0ea62d5e_2_1">
False
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<span id="answer_356fa8a9fa894a6db8a9717b0ea62d5e_2_1"/>
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<span class="status unanswered" id="status_356fa8a9fa894a6db8a9717b0ea62d5e_2_1" data-tooltip="Not yet answered.">
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</div>
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<span id="solution_356fa8a9fa894a6db8a9717b0ea62d5e_solution_1"/>
</div></div>
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<input type="hidden" name="problem_id" value="PART C" />
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<h3 class="hd hd-3 problem-header" id="03dbfb4cde7e4b29a6fa6ae7390c8409-problem-title" aria-describedby="block-v1:MITx+15.053x+3T2016+type@problem+block@03dbfb4cde7e4b29a6fa6ae7390c8409-problem-progress" tabindex="-1">
PART D
</h3>
<div class="problem-progress" id="block-v1:MITx+15.053x+3T2016+type@problem+block@03dbfb4cde7e4b29a6fa6ae7390c8409-problem-progress"></div>
<div class="problem">
<div><p>The following feasible region described is convex:</p>
\(\{(x,y): (x,y) \neq (0,0) \}\)
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><div class="choicegroup capa_inputtype" id="inputtype_03dbfb4cde7e4b29a6fa6ae7390c8409_2_1">
<fieldset aria-describedby="status_03dbfb4cde7e4b29a6fa6ae7390c8409_2_1">
<div class="field">
<input type="radio" name="input_03dbfb4cde7e4b29a6fa6ae7390c8409_2_1" id="input_03dbfb4cde7e4b29a6fa6ae7390c8409_2_1_choice_1a#A" class="field-input input-radio" value="choice_1a#A"/><label id="03dbfb4cde7e4b29a6fa6ae7390c8409_2_1-choice_1a#A-label" for="input_03dbfb4cde7e4b29a6fa6ae7390c8409_2_1_choice_1a#A" class="response-label field-label label-inline" aria-describedby="status_03dbfb4cde7e4b29a6fa6ae7390c8409_2_1">
True
</label>
</div>
<div class="field">
<input type="radio" name="input_03dbfb4cde7e4b29a6fa6ae7390c8409_2_1" id="input_03dbfb4cde7e4b29a6fa6ae7390c8409_2_1_choice_1a#B" class="field-input input-radio" value="choice_1a#B"/><label id="03dbfb4cde7e4b29a6fa6ae7390c8409_2_1-choice_1a#B-label" for="input_03dbfb4cde7e4b29a6fa6ae7390c8409_2_1_choice_1a#B" class="response-label field-label label-inline" aria-describedby="status_03dbfb4cde7e4b29a6fa6ae7390c8409_2_1">
False
</label>
</div>
<span id="answer_03dbfb4cde7e4b29a6fa6ae7390c8409_2_1"/>
</fieldset>
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<span class="status unanswered" id="status_03dbfb4cde7e4b29a6fa6ae7390c8409_2_1" data-tooltip="Not yet answered.">
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</span>
</div>
</div></div>
<div class="solution-span">
<span id="solution_03dbfb4cde7e4b29a6fa6ae7390c8409_solution_1"/>
</div></div>
<div class="action">
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PART E
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<p>The feasible region described by the following graph is convex:</p>
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