Techniques of Differentiation - Product Rule

The diagram below shows the graph of \(\mathrm{f}(x)=x^2e^{-x}\) for \(0 \le x \le 6\). There are points of inflexion at \(A\) and \(C\) and there is a maximum at \(B\).

Graph of f(x) = x^2 * e^-x

1. Find \(\mathrm{f}'(x)\).
2. Find the exact value of the \(y\)-coordinate of \(B\).
3. Sub-questions:
   1. Show that \(\mathrm{f}''(x) = (x^2 - 4x + 2)e^{-x}\).
   2. Hence, find the exact value of the \(x\)-coordinate of \(C\).

Diagram correction: the dashed height marked 0.5 is only schematic. The exact height of \(B\) is to be found in part (b); it is not exactly 0.5.