learnmath.bernatchez.net

Consider \(f(x) = x\ln(4 - x^2)\), for \(-2 < x < 2\).

Part of the graph of \(f\) is given below.

Let \(P\) and \(Q\) be the points on the curve of \(f\) where the tangent to the graph of \(f\) is parallel to the x-axis.

1. 1. Find the x-coordinate of \(P\) and \(Q\).

   2. Consider \(f(x) = k\).

      Write down all values of \(k\) for which there are exactly two solutions.

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

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

3. Sketch the graph of \(g^\prime\).

4. Consider \(g^\prime(x) = w\).

   Write down all values of \(w\) for which there are exactly two solutions.