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<h2 class="hd hd-2 unit-title">2. Graphing and Critical Points</h2>
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<p><b class="bfseries">Objectives</b></p><p>
At the end of this sequence, and after some practice, you should be able to: </p><ul class="itemize"><li><p>
Determine the shape of a function's graph based on information from its derivatives. </p></li><li><p>
Classify critical points using the First and Second Derivative Tests. </p></li></ul><p><b class="bfseries">Contents: 17 pages</b></p><p>
8 videos (29 minutes 1x speed) 23 questions </p>
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<h2 class="hd hd-2 unit-title">3. Preliminaries</h2>
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Given the same function [mathjaxinline]\ f(x) = 2x - 4\sin x[/mathjaxinline], which of the following pictures is most likely to be the graph of [mathjaxinline]\ f[/mathjaxinline] over [mathjaxinline][0,2\pi ][/mathjaxinline]? </p>
<center><img alt="The function f of x starts at the origin and is always positive. The function increases then decreases then increases. The function starts as concave down and then becomes concave up. " src="/assets/courseware/v1/447dfd5a95428696ca4d0ae6367a7088/asset-v1:MITx+18.01.1x+2T2019+type@asset+block/images_u4app1_sin1.svg" style="margin: 10px 25px 25px 25px" width="180px"/> &#8195;&#8195;&#8195;&#8195;<img alt="The function f of x starts at the origin and starts as negative and decreasing. Then f of x increases (becoming positive) and then decreases. The function starts as concave up and then becomes concave down." src="/assets/courseware/v1/82249dbb03222f573fde8e7e9be74a4c/asset-v1:MITx+18.01.1x+2T2019+type@asset+block/images_u4app1_sin2.svg" style="margin: 10px 25px 25px 25px" width="180px"/> &#8195;&#8195;&#8195;&#8195;<img alt="The function f of x starts at the origin and starts as negative and decreasing. Then f of x increases (becoming positive), then decreases, and then increases again. The function starts as concave up, then becomes concave down, then becomes concave up again." src="/assets/courseware/v1/8eb304243fab536a9ac1e02afe6e3927/asset-v1:MITx+18.01.1x+2T2019+type@asset+block/images_u4app1_sin3.svg" style="margin: 10px 25px 25px 25px" width="180px"/> &#8195;&#8195;&#8195;&#8195;<br/><br/><img alt="The function f of x starts at the origin and is always positive. The function is increasing, then decreasing, then increasing, then decreasing. The function starts as concave down, then becomes concave up, and then becomes concave down again." src="/assets/courseware/v1/a2771dc9fb302a4c6ae10a9018dde196/asset-v1:MITx+18.01.1x+2T2019+type@asset+block/images_u4app1_sin4.svg" style="margin: 10px 25px 25px 25px" width="180px"/> &#8195;&#8195;&#8195;&#8195;<img alt="The function f of x starts at the origin and is always positive and always increasing. The function starts as concave up and then becomes concave down." src="/assets/courseware/v1/77cc9c0cd4b5d06d78f14c22993a47fe/asset-v1:MITx+18.01.1x+2T2019+type@asset+block/images_u4app1_sin5.svg" style="margin: 10px 25px 25px 25px" width="180px"/> </center>
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If the graph of a function [mathjaxinline]\ f[/mathjaxinline] changes from increasing to decreasing, or vice versa, at the point [mathjaxinline]x=3[/mathjaxinline], which of the following can be true? Select as many as are possible. </p>
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<h2 class="hd hd-2 unit-title">4. Critical points</h2>
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The critical points of a function [mathjaxinline]\ f(x)[/mathjaxinline] to be all points [mathjaxinline]x[/mathjaxinline] in the domain of [mathjaxinline]f(x)[/mathjaxinline] such that </p><ul class="itemize"><li><p>
[mathjaxinline]f'(x) = 0[/mathjaxinline], or </p></li><li><p>
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Classifying Critical Points
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<img alt="At the point A, the function changes from increasing to decreasing and has a horizontal tangent line. At the point B, the function crosses the x axis. At the point C, the function changes from decreasing to increasing and has a sharp corner. At the point D, the function is increasing, changes from concave down to concave up, has a horizontal tangent line. At the piont E, The function changes from decreasing to increasing and has a horizontal tangent line." src="/assets/courseware/v1/59e98ea5572d715c6b5bd5cf83534c78/asset-v1:MITx+18.01.1x+2T2019+type@asset+block/images_u4app1_extrema.svg" style="margin: 10px 25px 25px 25px" width="400px"/>
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<p>
Choose all points on the graph that are </p>
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<text>A</text>
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<text>B</text>
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<text>C</text>
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<text>D</text>
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<text>E</text>
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</td>
<td style="text-align:right; border:none">
&#8195;&#8195;&#8195;&#8195;</td>
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<p style="display:inline">&#8195;local maxima:</p>
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<text>A</text>
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<text>C</text>
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<text>D</text>
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<text>E</text>
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&#8195;&#8195;&#8195;&#8195;</td>
<td style="text-align:left; border:none"><p style="display:inline">&#8195;local minima:</p> &#8195;&#8195;&#8195;&#8195;<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 3" role="group"><div class="choicegroup capa_inputtype" id="inputtype_app_1-tab4-problem1_4_1">
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Calculate critical points 1
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Find all [mathjaxinline]x[/mathjaxinline]-values of points that are critical points of [mathjaxinline]g(x) = 0.2x^5 + 2x^4 + 4x^3 - 17[/mathjaxinline]. </p>
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(Separate multiple points with a comma; e.g. 1.1, 2.3, 4. If there aren't any, type none.) </p>
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Find all [mathjaxinline]x[/mathjaxinline]-values of points that are critical points of [mathjaxinline]h(x) = 1.5 x^{2/3} + x[/mathjaxinline]. </p>
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(Separate multiple points with a comma; e.g. 1.1, 2.3, 4. If there aren't any, type none.) </p>
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We will see this function again, so you may wish to save your calculations! </p>
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<h2 class="hd hd-2 unit-title">5. First Derivative Test</h2>
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Determining the sign of the derivative
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<p>
Let's imagine that we have a continuous function [mathjaxinline]\ f[/mathjaxinline]. We've calculated its derivative [mathjaxinline]\ f'[/mathjaxinline]; the <span style="color:#99182C"><b class="bf">only</b></span> three critical points of [mathjaxinline]\ f[/mathjaxinline] are at [mathjaxinline]x = 0,[/mathjaxinline] [mathjaxinline]x=4[/mathjaxinline] and [mathjaxinline]x = 8[/mathjaxinline]. </p>
<p>
<span style="color:#99182C">
<b class="bf">Assume [mathjaxinline]\ f'[/mathjaxinline] is continuous at all other points. </b>
</span>
</p>
<center>
<img alt="Number line with x equals 0, x equals 4, and x equals 8 labeled." src="/assets/courseware/v1/a8ba00392798a3d76daacdda5a87c0b3/asset-v1:MITx+18.01.1x+2T2019+type@asset+block/images_u4app1_numline.svg" style="margin: 10px 25px 25px 25px" width="400px"/>
</center>
<p>
Finally, suppose that [mathjaxinline]\ f'(6)&gt;0[/mathjaxinline]. What (if anything) can we say about the following values of [mathjaxinline]\ f'[/mathjaxinline]? </p>
<p>
<p style="display:inline">[mathjaxinline]\ f'(-2)[/mathjaxinline]: </p>
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<option value="Must be positive."> Must be positive.</option>
<option value="Must be negative."> Must be negative.</option>
<option value="We cannot determine the sign."> We cannot determine the sign.</option>
</select>
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<span class="status unanswered" id="status_app_1-tab5-problem1_2_1" data-tooltip="Not yet answered.">
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<br/>
<p style="display:inline">[mathjaxinline]\ f'(2)[/mathjaxinline]: </p>
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<select name="input_app_1-tab5-problem1_3_1" id="input_app_1-tab5-problem1_3_1" aria-describedby="status_app_1-tab5-problem1_3_1">
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<option value="Must be positive."> Must be positive.</option>
<option value="Must be negative."> Must be negative.</option>
<option value="We cannot determine the sign."> We cannot determine the sign.</option>
</select>
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<br/>
<p style="display:inline">[mathjaxinline]\ f'(5)[/mathjaxinline]: </p>
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<option value="We cannot determine the sign."> We cannot determine the sign.</option>
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<br/>
<p style="display:inline">[mathjaxinline]\ f'(7)[/mathjaxinline]: </p>
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<option value="We cannot determine the sign."> We cannot determine the sign.</option>
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Changing the Function
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<p>
Suppose [mathjaxinline]\ f[/mathjaxinline] has a local maximum at [mathjaxinline]x=3[/mathjaxinline]. </p>
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Define a function [mathjaxinline]g(x)[/mathjaxinline] and an interval [mathjaxinline]I[/mathjaxinline] such that </p>
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[mathjaxinline]g(x) = f(x)[/mathjaxinline] for [mathjaxinline]x[/mathjaxinline] in [mathjaxinline]I[/mathjaxinline], </p>
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but [mathjaxinline]g(x) \ne f(x)[/mathjaxinline] for [mathjaxinline]x[/mathjaxinline] not in [mathjaxinline]I[/mathjaxinline]. </p>
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Identify all of the intervals [mathjaxinline]I[/mathjaxinline] defined below for which [mathjaxinline]g[/mathjaxinline] must have a local maximum at [mathjaxinline]x=3[/mathjaxinline]. </p>
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<h2 class="hd hd-2 unit-title">7. The First Derivative Test</h2>
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<p><h3>Finding Local Maxima and Minima</h3></p><p>
Suppose the function [mathjaxinline]\ f(x)[/mathjaxinline] is continuous at [mathjaxinline]x=a[/mathjaxinline] and has a critical point at [mathjaxinline]x=a[/mathjaxinline]. </p><table class="tabular" cellspacing="0" style="table-layout:auto"><tr><td style="text-align:left; border:none">
[mathjaxinline]\ f[/mathjaxinline] has a local minimum at [mathjaxinline]x=a[/mathjaxinline] if [mathjaxinline]\ f'(x) < 0[/mathjaxinline] just to the left of [mathjaxinline]a[/mathjaxinline] and [mathjaxinline]\ f'(x) > 0[/mathjaxinline] just to the right of [mathjaxinline]a[/mathjaxinline]. </td><td style="text-align:center; border:none"><img src="/assets/courseware/v1/1e03b0de4b279c259dca7ea419ddd2c5/asset-v1:MITx+18.01.1x+2T2019+type@asset+block/images_u4app1_test2.svg" width="100px" alt="See text on the left." style="margin: 10px 25px 25px 25px"/></td></tr><tr><td style="text-align:left; border:none">
[mathjaxinline]\ f[/mathjaxinline] has a local maximum at [mathjaxinline]x=a[/mathjaxinline] if [mathjaxinline]\ f'(x) > 0[/mathjaxinline] just to the left of [mathjaxinline]a[/mathjaxinline] and [mathjaxinline]\ f'(x) < 0[/mathjaxinline] just to the right of [mathjaxinline]a[/mathjaxinline]. </td><td style="text-align:center; border:none"><img src="/assets/courseware/v1/c777088002a0e851a6d08b7b0bb00da2/asset-v1:MITx+18.01.1x+2T2019+type@asset+block/images_u4app1_test1.svg" width="100px" alt="See text on the left." style="margin: 10px 25px 25px 25px"/></td></tr><tr><td style="text-align:left; border:none">
The point [mathjaxinline]x=a[/mathjaxinline] is neither a local minimum nor a local maximum of [mathjaxinline]\ f[/mathjaxinline] if [mathjaxinline]\ f'(x)[/mathjaxinline] has the same sign just to the left of [mathjaxinline]a[/mathjaxinline] and just to the right of [mathjaxinline]a[/mathjaxinline]. </td><td style="text-align:center; border:none"><img src="/assets/courseware/v1/d1be24440af85083dcb862e2dd21c119/asset-v1:MITx+18.01.1x+2T2019+type@asset+block/images_u4app1_test3.svg" width="100px" alt="See text on the left." style="margin: 10px 25px 25px 25px"/></td></tr></table><p><div class="hideshowbox"><h4 onclick="hideshow(this);" style="margin: 0px">Just to the left or right<span class="icon-caret-down toggleimage"/></h4><div class="hideshowcontent"><p>
When we use the phrase “[mathjaxinline]\ f'(x)>0[/mathjaxinline] just to the left of [mathjaxinline]a[/mathjaxinline]," we mean that there is some open interval [mathjaxinline](a-c,a)[/mathjaxinline] of positive width [mathjaxinline]c[/mathjaxinline] on which [mathjaxinline]\ f'[/mathjaxinline] is positive. This interval does not have to be very big, as long as it has some size! </p><p>
Similarly, “[mathjaxinline]\ f'(x)>0[/mathjaxinline] just to the right of [mathjaxinline]a[/mathjaxinline]" means that there is some open interval [mathjaxinline](a, a+d)[/mathjaxinline] of positive width [mathjaxinline]d[/mathjaxinline] on which [mathjaxinline]\ f'[/mathjaxinline] is positive. </p></div><p class="hideshowbottom" onclick="hideshow(this);" style="margin: 0px"><a href="javascript: {return false;}">Show</a></p></div></p><SCRIPT src="/assets/courseware/v1/631e447105fca1b243137b21b9ed6f90/asset-v1:MITx+18.01.1x+2T2019+type@asset+block/latex2edx.js" type="text/javascript"/><LINK href="/assets/courseware/v1/daf81af0af57b85a105e0ed27b7873a0/asset-v1:MITx+18.01.1x+2T2019+type@asset+block/latex2edx.css" rel="stylesheet" type="text/css"/>
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<h2 class="hd hd-2 unit-title">8. Applying the First Derivative Test</h2>
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Classify Critical Points 1
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List all [mathjaxinline]x[/mathjaxinline]-values of points at which the function [mathjaxinline]g(x) = x^5/5 + 2x^4 + 4x^3 - 17[/mathjaxinline] has a local maximum. </p>
<p>
(Separate multiple points with a comma; e.g. 2.1 , 3. If there aren't any, type none.) </p>
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<p>
List all points at which [mathjaxinline]g[/mathjaxinline] has a local minimum. </p>
<p>
(Separate multiple points with a comma; e.g. 2.1 , 3. If there aren't any, type none.) </p>
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Classify Critical Points 2
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List all [mathjaxinline]x[/mathjaxinline]-values of points at which the function [mathjaxinline]h(x) = 1.5 x^{2/3} + x[/mathjaxinline] has a local maximum. </p>
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(Separate multiple points with a comma. If there aren't any, type "none".) </p>
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<p>
List all points at which [mathjaxinline]h[/mathjaxinline] has a local minimum. </p>
<p>
(Separate multiple points with a comma. If there aren't any, type "none".) </p>
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<h2 class="hd hd-2 unit-title">9. Beyond the First Derivative Test</h2>
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Consider the function [mathjaxinline]\ f(x) = x^2 + 3\cos x[/mathjaxinline]. We know that its derivative is [mathjaxinline]\ f'(x) = 2x - 3\sin x[/mathjaxinline], so [mathjaxinline]\ f'(0) = 0[/mathjaxinline]. Thus, we have a critical point at [mathjaxinline]x=0[/mathjaxinline] and we want to know whether it is a local maximum or local minimum. </p>
<center>
<img alt="Number line with x equals 0 labeled." src="/assets/courseware/v1/c2f3414a7bc2e9f44629e73e275068e4/asset-v1:MITx+18.01.1x+2T2019+type@asset+block/images_u4app1_numline2.svg" style="margin: 10px 25px 25px 25px" width="400px"/>
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Using <b class="bfseries">only</b> the following information (i.e. do not use the formula given above) </p>
<ul class="itemize">
<li>
<p>
[mathjaxinline]\ f'(0) = 0[/mathjaxinline] </p>
</li>
<li>
<p>
[mathjaxinline]\ f'(\pi /2) = \pi - 3 &gt; 0[/mathjaxinline] </p>
</li>
<li>
<p>
[mathjaxinline]\ f'(-\pi /2) = -\pi + 3 &lt; 0[/mathjaxinline], </p>
</li>
</ul>
<p>
what can we conclude about the behavior of [mathjaxinline]\ f[/mathjaxinline] at [mathjaxinline]x=0[/mathjaxinline]? </p>
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<text> [mathjaxinline]\ f[/mathjaxinline] has a local maximum at [mathjaxinline]x=0[/mathjaxinline]</text>
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<h3 class="hd hd-2">The Second Derivative Test</h3>
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<h2 class="hd hd-2 unit-title">10. Finishing the Second Derivative Test</h2>
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Finishing the Second Derivative Test 1
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Let [mathjaxinline]g(x) = x^3[/mathjaxinline] and [mathjaxinline]h(x) = x^4[/mathjaxinline]. Then both [mathjaxinline]g'(0) = h'(0) = 0[/mathjaxinline], so both functions have a critical point at [mathjaxinline]x=0[/mathjaxinline]. What are their second derivatives at [mathjaxinline]x=0[/mathjaxinline]? </p>
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<p style="display:inline">[mathjaxinline]g^{\prime \prime }(0) =[/mathjaxinline] </p>
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Finishing the Second Derivative Test 2
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This is the borderline case for the Second Derivative Test. Now we want to figure out whether these critical points at [mathjaxinline]x=0[/mathjaxinline] are local extrema. </p>
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<p style="display:inline">When [mathjaxinline]x&lt;0[/mathjaxinline], &#8195;[mathjaxinline]g(x)=x^3[/mathjaxinline] is </p>
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<p style="display:inline">At [mathjaxinline]x=0[/mathjaxinline], &#8195;[mathjaxinline]g(x)=x^3[/mathjaxinline] has </p>
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<p style="display:inline">At [mathjaxinline]x=0[/mathjaxinline], &#8195;[mathjaxinline]h(x)=x^4[/mathjaxinline] has </p>
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Finishing the Second Derivative Test 3
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Given the answers to the previous questions, what is the best way to complete the Second Derivative Test in general? </p>
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If [mathjaxinline]\ f'(a) = 0[/mathjaxinline], and [mathjaxinline]\ f^{\prime \prime }(a) = 0[/mathjaxinline], then... </p>
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<text> [mathjaxinline]a[/mathjaxinline] is neither a local max nor a local min</text>
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<text> we do not have enough information to tell; the test is inconclusive.</text>
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<h2 class="hd hd-2 unit-title">11. The Second Derivative Test</h2>
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Suppose that [mathjaxinline]x=a[/mathjaxinline] is a critical point of [mathjaxinline]\ f[/mathjaxinline], with [mathjaxinline]\ f'(a) = 0[/mathjaxinline]. </p><table class="tabular" cellspacing="0" style="table-layout:auto"><tr><td style="text-align:left; border:none">
If [mathjaxinline]\ f^{\prime \prime }(a) > 0[/mathjaxinline], then [mathjaxinline]\ f[/mathjaxinline] has a local minimum at [mathjaxinline]x=a[/mathjaxinline].</td><td style="text-align:left; border:none"><img src="/assets/courseware/v1/a8b74dac86ee612f810857ee664cd3f1/asset-v1:MITx+18.01.1x+2T2019+type@asset+block/images_u4app1_test4.svg" width="200px" alt="See text on the left." style="margin: 10px 25px 25px 25px"/></td></tr><tr><td style="text-align:left; border:none">
If [mathjaxinline]\ f^{\prime \prime }(a) < 0[/mathjaxinline], then [mathjaxinline]\ f[/mathjaxinline] has a local maximum at [mathjaxinline]x=a[/mathjaxinline].</td><td style="text-align:left; border:none"><img src="/assets/courseware/v1/0986206230460827ea0d348145be62de/asset-v1:MITx+18.01.1x+2T2019+type@asset+block/images_u4app1_test5.svg" width="200px" alt="See text on the left." style="margin: 10px 25px 25px 25px"/></td></tr><tr><td style="text-align:left; border:none">
If [mathjaxinline]\ f^{\prime \prime }(a) = 0[/mathjaxinline], or does not exist, then the test is inconclusive — there might be a local maximum, or a local minimum, or neither. </td><td style="text-align:left; border:none"> </td></tr></table>
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<h2 class="hd hd-2 unit-title">12. Second Derivative Test practice</h2>
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Second Derivative Test practice
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Let [mathjaxinline]\ f(x) = e^ x - x - 2x^2.[/mathjaxinline] There is a critical point at [mathjaxinline]x=0[/mathjaxinline]; is it a local maximum or local minimum, or neither? </p>
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<p>
We've developed two tests for determining whether critical points are local minima or maxima. Each has its pros and cons. </p><ul class="itemize"><li><p>
The Second Derivative Test requires just one value of [mathjaxinline]\ f^{\prime \prime }[/mathjaxinline], but it is sometimes inconclusive. </p></li><li><p>
The First Derivative Test requires a bit more data, but is often able to provide more information as a result. </p></li></ul><p>
It is important to be comfortable using both tests! </p>
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<h2 class="hd hd-2 unit-title">13. Inflection Points</h2>
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An <span style="color:#27408C"><b class="bf">inflection point</b></span> is a point where the concavity of the function changes. That is the second derivative [mathjaxinline]f^{\prime \prime }(x)[/mathjaxinline] changes sign— [mathjaxinline]f^{\prime \prime }(x) > 0[/mathjaxinline] just to the left of [mathjaxinline]x[/mathjaxinline] and [mathjaxinline]f^{\prime \prime }(x) < 0[/mathjaxinline] just to the right of [mathjaxinline]x[/mathjaxinline] (or vice versa). </p>
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Finding Inflection Points
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Let [mathjaxinline]\ f(x) = 2x^6/5 - x^4/4 + 7x - 2/3[/mathjaxinline]. List all of the [mathjaxinline]x[/mathjaxinline]-values of inflection points for the function [mathjaxinline]\ f[/mathjaxinline]. </p>
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<h2 class="hd hd-2 unit-title">14. Graphing and the Second Derivative</h2>
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Using the Second Derivative
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<p>
Suppose that we know that [mathjaxinline]\ f'(x)[/mathjaxinline] is zero at [mathjaxinline]x=-2[/mathjaxinline] and [mathjaxinline]x=2[/mathjaxinline], is positive for [mathjaxinline]x&lt;-2[/mathjaxinline] and [mathjaxinline]x&gt;2[/mathjaxinline], and negative for [mathjaxinline]-2&lt;x&lt;2[/mathjaxinline]. </p>
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<p>
Suppose also that we know that [mathjaxinline]\ f^{\prime \prime }(x)[/mathjaxinline] is zero at [mathjaxinline]x=0[/mathjaxinline], [mathjaxinline]x=2[/mathjaxinline], and [mathjaxinline]x=4[/mathjaxinline], is positive on the intervals [mathjaxinline]0&lt;x&lt;2[/mathjaxinline] and [mathjaxinline]2&lt;x&lt;4[/mathjaxinline], and is negative for [mathjaxinline]x&lt;0[/mathjaxinline] or [mathjaxinline]x&gt;4[/mathjaxinline]. </p>
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<p>
List the [mathjaxinline]x-[/mathjaxinline]values of points where there is a local minimum. </p>
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(Separate multiple points with a comma. If there aren't any, type "none".) </p>
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List the [mathjaxinline]x-[/mathjaxinline]values of points where there is a local maximum. </p>
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List the [mathjaxinline]x-[/mathjaxinline]values of points where there is an inflection point. </p>
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(Separate multiple points with a comma. If there aren't any, type "none".) </p>
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(Choose the Extrema tool to label local max and min; choose the Inflection Point tool to label inflection points, and draw the function using the Function tool.) </p>
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Where are inflection points?
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True or false: If [mathjaxinline]\ f^{\prime \prime }(a) = 0[/mathjaxinline], then [mathjaxinline]a[/mathjaxinline] must be an inflection point for the function [mathjaxinline]\ f[/mathjaxinline]. <div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><div class="inputtype option-input ">
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<h2 class="hd hd-2 unit-title">15. Limits at Infinity</h2>
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Going to Infinity 1
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Suppose that a function [mathjaxinline]\ f[/mathjaxinline] is defined and continuous for all real numbers [mathjaxinline]x[/mathjaxinline], and </p>
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[mathjaxinline]\ f(2) = 1[/mathjaxinline], </p>
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[mathjaxinline]\ f'(x) &gt; 0[/mathjaxinline] for all [mathjaxinline]x\ge 2[/mathjaxinline], and </p>
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<p>
[mathjaxinline]\ f^{\prime \prime } (x) &gt; 0[/mathjaxinline] for all [mathjaxinline]x\ge 2[/mathjaxinline]. </p>
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Is there a point [mathjaxinline]x=a[/mathjaxinline] where [mathjaxinline]\ f(a) = 10000[/mathjaxinline]? </p>
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Going to Infinity 2
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Suppose that [mathjaxinline]\ g[/mathjaxinline] is defined and continuous for all real numbers [mathjaxinline]x[/mathjaxinline], and </p>
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<li>
<p>
[mathjaxinline]\ g(2) = 1[/mathjaxinline], </p>
</li>
<li>
<p>
[mathjaxinline]\ g'(x) &gt; 0[/mathjaxinline] for all [mathjaxinline]x\ge 2[/mathjaxinline], and </p>
</li>
<li>
<p>
[mathjaxinline]\ g^{\prime \prime } (x) &lt; 0[/mathjaxinline] for all [mathjaxinline]x\ge 2[/mathjaxinline]. </p>
</li>
</ul>
<p>
Is there a point [mathjaxinline]x=a[/mathjaxinline] where [mathjaxinline]g(a) = 10000[/mathjaxinline]? </p>
<p>
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<h2 class="hd hd-2 unit-title">16. The Third Derivative (Optional)</h2>
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Using the Third Derivative (Optional)
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In cases where the second derivative test is inconclusive, we can try to use the third derivative to determine whether a critical point is a local minimum or maximum. </p>
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Suppose that we have a critical point [mathjaxinline]x=a[/mathjaxinline] where [mathjaxinline]\ f'(a) = 0[/mathjaxinline] and [mathjaxinline]\ f^{\prime \prime }(a) = 0[/mathjaxinline] (so the second derivative test tells us nothing). Now suppose that [mathjaxinline]\ f^{\prime \prime \prime }(a) &gt; 0[/mathjaxinline]. </p>
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What can we conclude about [mathjaxinline]\ f^{\prime \prime }(x)[/mathjaxinline] when [mathjaxinline]x[/mathjaxinline] is just to the right of [mathjaxinline]a[/mathjaxinline]? <div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><div class="inputtype option-input ">
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Using the Third Derivative 2
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Remember that we're assuming that [mathjaxinline]\ f'(a) = 0[/mathjaxinline]. Given what we discovered about [mathjaxinline]\ f^{\prime \prime }[/mathjaxinline] just to the right of [mathjaxinline]x=a[/mathjaxinline], what can we conclude about [mathjaxinline]\ f'(x)[/mathjaxinline] when [mathjaxinline]x[/mathjaxinline] is just to the right of [mathjaxinline]a[/mathjaxinline]? <div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><div class="inputtype option-input ">
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Given what we discovered about [mathjaxinline]\ f^{\prime \prime }[/mathjaxinline] just to the left of [mathjaxinline]x=a[/mathjaxinline], what can we conclude about [mathjaxinline]\ f'(x)[/mathjaxinline] when [mathjaxinline]x[/mathjaxinline] is just to the left of [mathjaxinline]a[/mathjaxinline]? <div class="wrapper-problem-response" tabindex="-1" aria-label="Question 2" role="group"><div class="inputtype option-input ">
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Final Answer
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Given what we discovered about [mathjaxinline]\ f'[/mathjaxinline] near [mathjaxinline]x=a[/mathjaxinline], what can we conclude about the critical point? </p>
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<h2 class="hd hd-2 unit-title">17. Summary</h2>
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<p><b class="bfseries">Definition of Critical Points</b></p><p>
The critical points of a function [mathjaxinline]\ f(x)[/mathjaxinline] to be all points [mathjaxinline]x[/mathjaxinline] in the domain of [mathjaxinline]f(x)[/mathjaxinline] such that </p><ul class="itemize"><li><p>
[mathjaxinline]f'(x) = 0[/mathjaxinline], or </p></li><li><p>
[mathjaxinline]f'(x)[/mathjaxinline] does not exist. </p></li></ul><p><b class="bfseries">The First Derivative Test</b></p><p><b class="bfseries">Finding Local Maxima and Minima</b></p><p>
Suppose the function [mathjaxinline]\ f(x)[/mathjaxinline] is continuous at [mathjaxinline]x=a[/mathjaxinline] and has a critical point at [mathjaxinline]x=a[/mathjaxinline]. </p><table class="tabular" cellspacing="0" style="table-layout:auto"><tr><td style="text-align:left; border:none">
[mathjaxinline]\ f[/mathjaxinline] has a local minimum at [mathjaxinline]x=a[/mathjaxinline] if [mathjaxinline]\ f'(x) < 0[/mathjaxinline] just to the left of [mathjaxinline]a[/mathjaxinline] and [mathjaxinline]\ f'(x) > 0[/mathjaxinline] just to the right of [mathjaxinline]a[/mathjaxinline]. </td><td style="text-align:center; border:none"><img src="/assets/courseware/v1/1e03b0de4b279c259dca7ea419ddd2c5/asset-v1:MITx+18.01.1x+2T2019+type@asset+block/images_u4app1_test2.svg" width="100px" alt="See text on the left." style="margin: 10px 25px 25px 25px"/></td></tr><tr><td style="text-align:left; border:none">
[mathjaxinline]\ f[/mathjaxinline] has a local maximum at [mathjaxinline]x=a[/mathjaxinline] if [mathjaxinline]\ f'(x) > 0[/mathjaxinline] just to the left of [mathjaxinline]a[/mathjaxinline] and [mathjaxinline]\ f'(x) < 0[/mathjaxinline] just to the right of [mathjaxinline]a[/mathjaxinline]. </td><td style="text-align:center; border:none"><img src="/assets/courseware/v1/c777088002a0e851a6d08b7b0bb00da2/asset-v1:MITx+18.01.1x+2T2019+type@asset+block/images_u4app1_test1.svg" width="100px" alt="See text on the left" style="margin: 10px 25px 25px 25px"/></td></tr><tr><td style="text-align:left; border:none">
The point [mathjaxinline]x=a[/mathjaxinline] is neither a local minimum nor a local maximum of [mathjaxinline]\ f[/mathjaxinline] if [mathjaxinline]\ f'(x)[/mathjaxinline] has the same sign just to the left of [mathjaxinline]a[/mathjaxinline] and just to the right of [mathjaxinline]a[/mathjaxinline]. </td><td style="text-align:center; border:none"><img src="/assets/courseware/v1/d1be24440af85083dcb862e2dd21c119/asset-v1:MITx+18.01.1x+2T2019+type@asset+block/images_u4app1_test3.svg" width="100px" alt="See text on the left." style="margin: 10px 25px 25px 25px"/></td></tr></table><p><div class="hideshowbox"><h4 onclick="hideshow(this);" style="margin: 0px">Just to the left or right<span class="icon-caret-down toggleimage"/></h4><div class="hideshowcontent"><p>
When we use the phrase “[mathjaxinline]\ f'(x)>0[/mathjaxinline] just to the left of [mathjaxinline]a[/mathjaxinline]," we mean that there is some open interval [mathjaxinline](a-c,a)[/mathjaxinline] of positive width [mathjaxinline]c[/mathjaxinline] on which [mathjaxinline]\ f'[/mathjaxinline] is positive. This interval does not have to be very big, as long as it has some size! </p><p>
Similarly, “[mathjaxinline]\ f'(x)>0[/mathjaxinline] just to the right of [mathjaxinline]a[/mathjaxinline]" means that there is some open interval [mathjaxinline](a, a+d)[/mathjaxinline] of positive width [mathjaxinline]d[/mathjaxinline] on which [mathjaxinline]\ f'[/mathjaxinline] is positive. </p></div><p class="hideshowbottom" onclick="hideshow(this);" style="margin: 0px"><a href="javascript: {return false;}">Show</a></p></div></p><p><b class="bfseries">The Second Derivative Test</b></p><p>
Suppose that [mathjaxinline]x=a[/mathjaxinline] is a critical point of [mathjaxinline]\ f[/mathjaxinline], with [mathjaxinline]\ f'(a) = 0[/mathjaxinline]. </p><table class="tabular" cellspacing="0" style="table-layout:auto"><tr><td style="text-align:left; border:none">
If [mathjaxinline]\ f^{\prime \prime }(a) > 0[/mathjaxinline], then [mathjaxinline]\ f[/mathjaxinline] has a local minimum at [mathjaxinline]x=a[/mathjaxinline]. </td><td style="text-align:center; border:none"><img src="/assets/courseware/v1/a8b74dac86ee612f810857ee664cd3f1/asset-v1:MITx+18.01.1x+2T2019+type@asset+block/images_u4app1_test4.svg" width="100px" alt="See text on the left." style="margin: 10px 25px 25px 25px"/></td></tr><tr><td style="text-align:left; border:none">
If [mathjaxinline]\ f^{\prime \prime }(a) < 0[/mathjaxinline], then [mathjaxinline]\ f[/mathjaxinline] has a local maximum at [mathjaxinline]x=a[/mathjaxinline]. </td><td style="text-align:center; border:none"><img src="/assets/courseware/v1/0986206230460827ea0d348145be62de/asset-v1:MITx+18.01.1x+2T2019+type@asset+block/images_u4app1_test5.svg" width="100px" alt="See text on the left." style="margin: 10px 25px 25px 25px"/></td></tr><tr><td style="text-align:left; border:none">
If [mathjaxinline]\ f^{\prime \prime }(a) = 0[/mathjaxinline], or does not exist, then the test is inconclusive — there might be a local maximum, or a local minimum, or neither. </td><td style="text-align:center; border:none"> </td></tr></table><p><b class="bfseries">The First Derivative Test vs. the Second Derivative Test</b></p><p>
We've developed two tests for determining whether critical points are local minima or maxima. Each has its pros and cons. </p><ul class="itemize"><li><p>
The Second Derivative Test requires just one value of [mathjaxinline]\ f^{\prime \prime }[/mathjaxinline], but it is sometimes inconclusive. </p></li><li><p>
The First Derivative Test requires a bit more data, but is often able to provide more information as a result. </p></li></ul><p>
It is important to be comfortable using both tests! </p><p>
An <span style="color:#27408C"><b class="bf">inflection point</b></span> is a point where the concavity of the function changes. That is the second derivative [mathjaxinline]f^{\prime \prime }(x)[/mathjaxinline] changes sign— [mathjaxinline]f^{\prime \prime }(x) > 0[/mathjaxinline] just to the left of [mathjaxinline]x[/mathjaxinline] and [mathjaxinline]f^{\prime \prime }(x) < 0[/mathjaxinline] just to the right of [mathjaxinline]x[/mathjaxinline] (or vice versa). </p><SCRIPT src="/assets/courseware/v1/631e447105fca1b243137b21b9ed6f90/asset-v1:MITx+18.01.1x+2T2019+type@asset+block/latex2edx.js" type="text/javascript"/><LINK href="/assets/courseware/v1/daf81af0af57b85a105e0ed27b7873a0/asset-v1:MITx+18.01.1x+2T2019+type@asset+block/latex2edx.css" rel="stylesheet" type="text/css"/>
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