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      <text>12.2.2.1 Distinguishing between local maxima and minima</text>
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      <text><![CDATA[<h3>Distinguishing between local maxima and minima</h3><br>
<p>We might ask if it is possible to predict when a stationary point is a local maximum, a local minimum or a point of inflection without the necessity of drawing the curve. To do this we highlight the general characteristics of curves in the neighbourhood
    of local maxima and minima.</p>
<p>For example: at a local maximum (located at \(x_0\) say) describes the situation: <br></p>
<figure class="HELM_figure"><img src="@@PLUGINFILE@@/12.2%20figure%2011.svg" alt="parabola with x0 maximum" class="img-responsive atto_image_button_text-bottom" width="204" height="125"></figure>
<p></p>
<p>To the left of the maximum \(\dfrac{\mathrm{d}f}{\mathrm{d}x}&gt;0\)</p>
<p>To the right of the maximum \(\dfrac{\mathrm{d}f}{\mathrm{d}x}&lt;0\)</p><br>
<p></p>
<p>If we draw a graph of the <strong>derivative</strong> \(\dfrac{\mathrm{d}f}{\mathrm{d}x}\) against \(x\) then, near a local maximum, it <strong>must</strong> take one of two basic shapes described:</p>
<figure class="HELM_figure"><img src="@@PLUGINFILE@@/12.2%20figure%2012.svg" alt="inflexions at x0 with angle between y equals 0 as alpha, second graph alpha is 180 degrees" class="img-responsive atto_image_button_text-bottom" width="409" height="132" style="font-size: 0.9375rem;"></figure>
<p>In case (a) \( \dfrac{\mathrm{d}}{\mathrm{d}x}\left(\dfrac{\mathrm{d}f}{\mathrm{d}x}\right)\bigg|_{x_0}\equiv \tan\alpha&lt;0\) whilst in case (b) \(\dfrac{\mathrm{d}}{\mathrm{d}x}\left(\dfrac{\mathrm{d}f}{\mathrm{d}x}\right)\bigg|_{x_0}=0\).</p>
<p>We reach the conclusion that at a stationary point which is a maximum the value of the second derivative \(\dfrac{\mathrm{d}^2f}{\mathrm{d} x^2}\) is either negative or zero.</p>
<p>Near a local minimum the above graphs are inverted. Below shows a local minimum. <br></p>
<p style="text-align: center;"><img src="@@PLUGINFILE@@/12.2%20figure%2013.svg" alt="parabola with minimum x0" class="img-responsive atto_image_button_text-bottom" width="199" height="116"></p>
<p style="text-align: left;">To the left of the minimum \(\dfrac{\mathrm{d}f}{\mathrm{d}x}&lt;0\)</p>
<p style="text-align: left;">To the right of the minimum \(\dfrac{\mathrm{d}f}{\mathrm{d}x}&gt;0\)<br></p>
<p>The figure below shows the two possible graphs of the derivative: <br></p>
<figure class="HELM_figure"><img src="@@PLUGINFILE@@/12.2%20figure%2014.svg" alt="two graphs with inflexion at x0 and angle between x is 0 is beta" class="img-responsive atto_image_button_text-bottom" width="328" height="92"></figure>
<p>Here, for case (a) \(\dfrac{\mathrm{d}}{\mathrm{d}x}\left(\dfrac{\mathrm{d}f}{\mathrm{d}x}\right)\bigg|_{x_0}=\tan\beta &gt;0\) whilst in (b) \( \dfrac{\mathrm{d}}{\mathrm{d}x}\left(\dfrac{\mathrm{d}f}{\mathrm{d}x}\right)\bigg|_{x_0}=0.\).</p>
<p>In this case we conclude that at a stationary point which is a minimum the value of the second derivative \(\dfrac{\mathrm{d} ^2f}{\mathrm{d} x^2}\) is either positive or zero. For the third possibility for a stationary point - a point of inflection - the graph of \(f(x)\)
    against \(x\) and of \(\dfrac{\mathrm{d}f}{\mathrm{d}x}\) against \(x\) take one of two forms as shown below.<br></p>
<figure class="HELM_figure"><img src="@@PLUGINFILE@@/12.2%20figure%2015.svg" alt="4 graphs describing the when df by dx is positive and negative." class="img-responsive atto_image_button_text-bottom" width="451" height="313" style="font-size: 0.9375rem;"></figure>
<p>For either of these cases \(\dfrac{\mathrm{d}}{\mathrm{d}x}\left(\dfrac{\mathrm{d}f}{\mathrm{d}x}\right)\bigg|_{x_0}=0\)</p>
<p>The sketches and analysis of the shape of a curve \(y=f(x)\) in the near neighbourhood of stationary points allow us to make the following important deduction:</p>
<p>A generalisation of the third law of indices states:</p>
<div class="HELM_keypoint">
    <h4>Key Point</h4>
    <p>If \(x_0\) locates a stationary point of \(f(x)\), so that \(\displaystyle \frac{\mathrm{d}f}{\mathrm{d}x}\|_{x_0}= 0\), then the stationary point<br></p>
    <p>\(\quad\) is a local minimum if \(\dfrac{\mathrm{d}^2f}{\mathrm{d}x^2}\bigg|_{x_0}&gt;0\) <br></p>
    <p>\(\quad\) is a local maximum if \(\dfrac{\mathrm{d}^2f}{\mathrm{d}x^2}\bigg|_{x_0}&lt;0\) <br></p>
    <p>\(\quad\) is inconclusive if \(\dfrac{\mathrm{d}^2f}{\mathrm{d}x^2}\bigg|_{x_0}=0\).<br>
    </p>
</div>
<hr>
<h4 class="HELM_example">Example</h4>
<p>Find the stationary points of the function \(f(x)=x^3-6x\). </p>
<p>Are these stationary points local maxima or local minima?</p>
<h4 class="HELM_solution">Solution</h4>
<p>\(\dfrac{\mathrm{d}f}{\mathrm{d}x}= 3x^2−6\). At a stationary point \(\dfrac{\mathrm{d}f}{\mathrm{d}x}= 0\) so \(3x^2−6 = 0\), implying \(x=\pm\sqrt{2}\).Thus \(f(x)\) has stationary points at \(x=\sqrt{2}\)and\(x=−\sqrt{2}\). To decide if these are maxima or minima we examine the value
    of the second derivative of \(f(x)\)at the stationary points. <br></p>
<p>\(\dfrac{\mathrm{d}^2f}{\mathrm{d}x^2}= 6x\) so \(\dfrac{\mathrm{d}^2f}{\mathrm{d}x^2}\bigg|_{x=\sqrt{2}}= 6\sqrt{2}&gt;0\). Hence \(x=\sqrt{2}\) locates a local minimum.</p>
<p>Similarly \(\dfrac{\mathrm{d}^2f}{\mathrm{d}x^2}\bigg|_{x=−\sqrt{2}}=−6\sqrt{2}&lt;0\). Hence \(x=−\sqrt{2}\) locates a local maximum.</p>
<p>A sketch of the curve agrees with this analysis:</p>
<figure class="HELM_figure"><img src="@@PLUGINFILE@@/12.2%20figure%2016.svg" alt="sin wave with maximum and minimum at minus square root 2 and square root 2." class="img-responsive atto_image_button_text-bottom" width="256" height="109"></figure>]]></text>
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