Derivatives of Exponential, Logarithmic and Trigonometric Functions

The following diagram shows the graph of \(f(x) = \mathrm{e}^{-x^2}\).

The points \(\mathrm{A}, \mathrm{B}, \mathrm{C}, \mathrm{D}\) and \(\mathrm{E}\) lie on the graph of \(f\). Two of these are points of inflexion.

1. Identify these \(\textbf{two}\) points of inflexion.[2]
2. 1. Find \(f'(x)\).
   2. Show that \(f''(x) = (4x^2 - 2)\mathrm{e}^{-x^2}\).[5]
3. Find the \(x\)-coordinate of each point of inflexion.[4]
4. Use the second derivative to show that one of these points is a point of inflexion.[4]