<p>The diagram below shows that the graph of \(y = \mathrm{g}'(x)\) has two \(x\)-intercepts at \((-1,0)\) and \((1,0)\) as well as two stationary points whose \(x\)-coordinates are \(-1\) and \(\frac{1}{2}\) respectively.</p>
<div class="content-block"><figure class="question-figure aligncenter" style="text-align: center;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/public/question-images/pasted/1779551605779-lhwen6spufr.png" style="max-width: min(280px, 100%); height: auto; display: block;" class="aligncenter custom-width" /></figure></div>
<ol class="i-list"><li><div class="list-item-paragraph">A stationary point on the curve of \(y = \mathrm{g}(x)\) has a \(x\)-coordinate that is positive. State the value of this \(x\)-coordinate and determine the nature of this stationary point on the curve \(y = \mathrm{g}(x)\).<span class="inline-marks">[2]</span></div></li></ol>
<p>It is given that&nbsp;\(\mathrm{g}'(x) = ax^4 + bx^3 + cx^2 + dx + e\).</p>
<ol class="i-list" start="2"><li><div class="list-item-paragraph">Show that&nbsp;\(a - b + c - d + e = 0\).<span class="inline-marks">[1]</span></div></li></ol>
<p>Given further that the gradient of the tangent at the point of inflexion on \(y = \mathrm{g}(x)\) is \(-\frac{27}{8}\), find the values of \(a\), \(b\), \(c\), \(d\) and \(e\).<span class="inline-marks">[4]</span></p>

The diagram below shows that the graph of \(y = \mathrm{g}'(x)\) has two \(x\)-intercepts at \((-1,0)\) and \((1,0)\) as well as two stationary points whose \(x\)-coordinates are \(-1\) and \(\frac{1}{2}\) respectively.

A stationary point on the curve of \(y = \mathrm{g}(x)\) has a \(x\)-coordinate that is positive. State the value of this \(x\)-coordinate and determine the nature of this stationary point on the curve \(y = \mathrm{g}(x)\).

It is given that \(\mathrm{g}'(x) = ax^4 + bx^3 + cx^2 + dx + e\).

Given further that the gradient of the tangent at the point of inflexion on \(y = \mathrm{g}(x)\) is \(-\frac{27}{8}\), find the values of \(a\), \(b\), \(c\), \(d\) and \(e\).

Applications of Differentiation

International Baccalaureate Higher Level Mathematics

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