Derivatives of Exponential, Logarithmic and Trigonometric Functions

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

1. 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.
   1. Find the \(x\)-coordinate of \(\mathrm{P}\) and of \(\mathrm{Q}\).
   2. Consider \(f(x) = k\). Write down all values of \(k\) for which there are exactly two solutions.[5]

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

2. Show that \(g'(x) = \frac{-2x^4}{4 - x^2} + 3x^2\ln{(4 - x^2)}\).[4]
3. Sketch the graph of \(g'\).[2]
4. Consider \(g'(x) = w\). Write down all values of \(w\) for which there are exactly two solutions.[3]