Extreme values on closed and bounded domains - Revision history

https://mathresearch.utsa.edu/wiki/index.php?action=history&feed=atom&title=Extreme_values_on_closed_and_bounded_domains Extreme values on closed and bounded domains - Revision history 2026-04-09T01:50:32Z Revision history for this page on the wiki MediaWiki 1.34.1 https://mathresearch.utsa.edu/wiki/index.php?title=Extreme_values_on_closed_and_bounded_domains&diff=3858&oldid=prev Khanh at 23:11, 14 November 2021 2021-11-14T23:11:49Z

<p></p> <table class="diff diff-contentalign-left" data-mw="interface"> <col class="diff-marker" /> <col class="diff-content" /> <col class="diff-marker" /> <col class="diff-content" /> <tr class="diff-title" lang="en"> <td colspan="2" style="background-color: #fff; color: #222; text-align: center;">← Older revision</td> <td colspan="2" style="background-color: #fff; color: #222; text-align: center;">Revision as of 23:11, 14 November 2021</td> </tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l117" >Line 117:</td> <td colspan="2" class="diff-lineno">Line 117:</td></tr> <tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div> Answer: point of inflection: <math> x = 0 \ </math>.</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div> Answer: point of inflection: <math> x = 0 \ </math>.</div></td></tr> <tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr> <tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>==<del class="diffchange diffchange-inline">Resources</del>==</div></td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>== <ins class="diffchange diffchange-inline">Licensing </ins>== </div></td></tr> <tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>* [https://en.wikibooks.org/wiki/Calculus/Extreme_Value_Theorem Extreme Value Theorem<del class="diffchange diffchange-inline">]</del>, Wikibooks: Calculus</div></td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins class="diffchange diffchange-inline">Content obtained and/or adapted from:</ins></div></td></tr> <tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>* [https://en.wikibooks.org/wiki/Calculus/Extreme_Value_Theorem Extreme Value Theorem, Wikibooks: Calculus<ins class="diffchange diffchange-inline">] under a CC BY-SA license</ins></div></td></tr> </table>

Khanh https://mathresearch.utsa.edu/wiki/index.php?title=Extreme_values_on_closed_and_bounded_domains&diff=2579&oldid=prev Lila at 17:41, 19 October 2021 2021-10-19T17:41:20Z

<p></p> <table class="diff diff-contentalign-left" data-mw="interface"> <col class="diff-marker" /> <col class="diff-content" /> <col class="diff-marker" /> <col class="diff-content" /> <tr class="diff-title" lang="en"> <td colspan="2" style="background-color: #fff; color: #222; text-align: center;">← Older revision</td> <td colspan="2" style="background-color: #fff; color: #222; text-align: center;">Revision as of 17:41, 19 October 2021</td> </tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l116" >Line 116:</td> <td colspan="2" class="diff-lineno">Line 116:</td></tr> <tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr> <tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div> Answer: point of inflection: <math> x = 0 \ </math>.</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div> Answer: point of inflection: <math> x = 0 \ </math>.</div></td></tr> <tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;"></ins></div></td></tr> <tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">==Resources==</ins></div></td></tr> <tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">* [https://en.wikibooks.org/wiki/Calculus/Extreme_Value_Theorem Extreme Value Theorem], Wikibooks: Calculus</ins></div></td></tr> </table>

Lila https://mathresearch.utsa.edu/wiki/index.php?title=Extreme_values_on_closed_and_bounded_domains&diff=2578&oldid=prev Lila: Created page with "<blockquote style="background: white; border: 1px solid black; padding: 1em;"> '''Extreme Value Theorem:''' If ''f'' is a continuous function and closed on the interval [<mat..." 2021-10-19T17:40:32Z

<p>Created page with "<blockquote style="background: white; border: 1px solid black; padding: 1em;"> '''Extreme Value Theorem:''' If ''f'' is a continuous function and closed on the interval [<mat..."</p> <p><b>New page</b></p><div><blockquote style="background: white; border: 1px solid black; padding: 1em;"> <br /> '''Extreme Value Theorem:'''<br /> If ''f'' is a continuous function and closed on the interval [<math>a,b</math>], then ''f'' has both a minimum and a maximum.</blockquote><br /> <br /> This introduces us to the aspect of global extrema and local extrema. (Also known as absolute extrema or relative extrema respectively.)<br /> <br /> How is this so? Let us use an example.<br /> <br /> <math> f(x) = x^2 </math> and is closed on the interval [-1,2]. Find all extrema.<br /> <br /> :<math> \frac{dy}{dx} = 2x</math><br /> <br /> A critical point (a point where the derivative is zero) exists at (0,0). Just for practice, let us use the second derivative test to evaluate whether or not it is a minimum or maximum. (You should know it is a minimum from looking at the graph.)<br /> <br /> :<math> \frac{d^2y}{dx^2} = 2</math><br /> <br /> <math> f''(c) > 0</math>, thus it must be a minimum.<br /> <br /> As mentioned before, one can find global extrema on a closed interval. How? Evaluate the ''y'' coordinate at the endpoints of the interval and compare it to the ''y'' coordinates of the critical point. When you are finding extrema on a closed interval it is called a local extremum and when it's for the whole graph it's called a global extremum.<br /> <br /> 1: Critical Point: (0,0) This is the lowest value in the interval. Therefore, it is a local minimum which also happens to be the global minimum.<br /> <br /> 2: Left Endpoint (-1, 1) This point is not a critical point nor is it the highest/lowest value, therefore it qualifies as nothing.<br /> <br /> 3: Right Endpoint (2, 4) This is the highest value in the interval, and thus it is a local maximum.<br /> <br /> This example was to show you the '''extreme value theorem'''. The quintessential point is this: on a closed interval, the function will have both minima and maxima. However, if that interval was an open interval of all real numbers, (0,0) would have been a local minimum. On a closed interval, always remember to evaluate endpoints to obtain global extrema.<br /> <br /> ==First Derivative Test==<br /> <br /> Recall that the first derivative of a function describes the slope of the graph of the function at every point along the graph for which the function is defined and differentiable.<br /> <br /> <blockquote style="background: white; border: 1px solid black; padding: 1em;"> <br /> '''Increasing/Decreasing:'''<br /> <br /> * If <math> f'(x) < 0 \ </math>, then <math> f(x) \ </math> is decreasing.<br /> <br /> * If <math> f'(x) > 0 \ </math>, then <math> f(x) \ </math> is increasing.<br /> <br /> '''Local Extrema:'''<br /> <br /> * If <math> \frac{dy}{dx}|_{x=c} = f'(c) = 0 </math> and <math> f'(x) \ </math> changes signs at <math> x = c \ </math>, then there exists a local extremum at <math> x = c \ </math>.<br /> <br /> * If <math> f'(x) < 0 \ </math> for <math> x < c \ </math> and <math> f'(x) > 0 \ </math> for <math> x > c \ </math>, then <math> f(c) \ </math> is a local minimum.<br /> <br /> * If <math> f'(x) > 0 \ </math> for <math> x < c \ </math> and <math> f'(x) < 0 \ </math> <math> x > c \ </math>, then <math> f(c) \ </math> is a local maximum.<br /> </blockquote><br /> <br /> '''Example 1:'''<br /> <br /> Let <math> f(x) = 3x^2 + 4x - 5 \ </math>. Find all local extrema.<br /> <br /> * Find <math> \frac{dy}{dx} </math><br /> <br /> : <math> f(x) = 3x^2 + 4x - 5 \ </math><br /> : <math> f'(x) = 6x + 4 \ </math><br /> <br /> * Set <math> \frac{dy}{dx} = 0 </math> to find local extrema.<br /> <br /> : <math> 6x + 4 = 0 \ </math><br /> : <math> 6x = -4 \ </math><br /> : <math> x = - \frac{2}{3} </math><br /> <br /> * Determine whether there is a local minimum or maximum at <math> x = - \frac{2}{3} </math>.<br /> <br /> : Choose an ''x'' value smaller than <math> - \frac{2}{3} </math>:<br /> <br /> : <math> f'(-1) = 6(-1) + 4 = -2 < 0 \ </math><br /> <br /> : Choose an ''x'' value larger than <math> - \frac{2}{3} </math>:<br /> <br /> : <math> f'(1) = 6(1) + 4 = 10 > 0 \ </math><br /> <br /> Therefore, there is a local minimum at <math> x = - \frac{2}{3} </math> because <math> f'(- \frac{2}{3}) = 0 </math> and <math> f'(x) \ </math> changes signs at <math> x = - \frac{2}{3} </math>.<br /> <br /> Answer: local minimum: <math> x = - \frac{2}{3} </math>.<br /> <br /> ==Second Derivative Test== <br /> <br /> Recall that the second derivative of a function describes the concavity of the graph of that function.<br /> <br /> <blockquote style="background: white; border: 1px solid black; padding: 1em;"> <br /> * If <math> \frac{d^2y}{dx^2}|_{x=c} = f''(c) = 0 </math> and <math> f''(c) \ </math> changes signs at <math> x = c \ </math>, then there is a point of inflection (change in concavity) at <math> x = c \ </math>.<br /> <br /> * If <math> f''(x) < 0 \ </math>, then the graph of <math> f(x) \ </math> is concave down.<br /> <br /> * If <math> f''(x) > 0 \ </math>, then the graph of <math> f(x) \ </math> is concave up.<br /> </blockquote><br /> <br /> '''Example 2:'''<br /> <br /> Let <math> f(x) = x^3 + 2x + 7 \ </math>. Find any points of inflection on the graph of <math> f(x) \ </math>.<br /> <br /> * Find <math> \frac{d^2y}{dx^2} </math>.<br /> <br /> : <math> f(x) = x^3 + 2x + 7 \ </math><br /> : <math> f'(x) = 3x^2 + 2 \ </math><br /> : <math> f''(x) = 6x \ </math><br /> <br /> * Set <math> \frac{d^2y}{dx^2} = 0 </math>.<br /> <br /> : <math> 6x = 0 \ </math><br /> : <math> x = 0 \ </math><br /> <br /> * Determine whether <math> f''(x) \ </math> changes signs at <math> x = 0 \ </math>.<br /> <br /> : Choose an ''x'' value that is smaller than 0:<br /> <br /> : <math> f''(-1) = 6(-1) = -6 < 0 \ </math><br /> <br /> : Choose an ''x'' value that is larger than 0:<br /> <br /> : <math> f''(1) = 6(1) = 6 > 0 \ </math><br /> <br /> Therefore, there exists a point of inflection at <math> x = 0 \ </math> because <math> f''(0) = 0 \ </math> and <math> f''(x) \ </math> changes signs at <math> x = 0 \ </math>.<br /> <br /> Answer: point of inflection: <math> x = 0 \ </math>.</div>

Lila