<p>Consider \(f(x) = x \ln{(4 - x^2)}\), for&nbsp;\(-2 &lt; x &lt; 2\). The graph of \(f\) is given below.</p>
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<ol class="abc-list"><li><div class="list-item-paragraph">Let&nbsp;\(\mathrm{P}\) and \(\mathrm{Q}\) be points pn  the curve of \(f\) where the tangent to the graph of \(f\) is parallel ti the \(x\)-axis.</div><ol class="i-list"><li><div class="list-item-paragraph">Find the \(x\)-coordinate of \(\mathrm{P}\) and of \(\mathrm{Q}\).</div></li><li><div class="list-item-paragraph">Consider \(f(x) = k\). Write down all values of \(k\) for which there are exactly two solutions.<span class="inline-marks">[5]</span></div></li></ol></li></ol>
<p>Let&nbsp;\(g(x) = x^3 \ln{(4 - x^2)}\), for&nbsp;\(-2 &lt; x&lt; 2\).</p>
<ol class="abc-list" start="2"><li><div class="list-item-paragraph">Show that&nbsp;\(g'(x) = \frac{-2x^4}{4 - x^2} + 3x^2\ln{(4 - x^2)}\).<span class="inline-marks">[4]</span></div></li><li><div class="list-item-paragraph">Sketch the graph of \(g'\).<span class="inline-marks">[2]</span></div></li><li><div class="list-item-paragraph">Consider \(g'(x) = w\). Write down all values of \(w\) for which there are exactly two solutions.<span class="inline-marks">[3]</span></div></li></ol>

Consider \(f(x) = x \ln{(4 - x^2)}\), for \(-2 < x < 2\). The graph of \(f\) is given below.

Let \(\mathrm{P}\) and \(\mathrm{Q}\) be points pn  the curve of \(f\) where the tangent to the graph of \(f\) is parallel ti the \(x\)-axis.

Let \(g(x) = x^3 \ln{(4 - x^2)}\), for \(-2 < x< 2\).

Show that \(g'(x) = \frac{-2x^4}{4 - x^2} + 3x^2\ln{(4 - x^2)}\).

Consider \(g'(x) = w\). Write down all values of \(w\) for which there are exactly two solutions.

Derivatives of Exponential, Logarithmic and Trigonometric Functions

International Baccalaureate Standard Level Mathematics