Determine parameter when the derivative vanishes

Problem

We are given the set of functions \(f_a : x\mapsto x\mathrm{e}^{ax}\) defined on \(\mathbb{R}\) with \(a\in\mathbb{R}\backslash\{0\}\). Determine the value of \(a\) for which the derivative of \(f_a\) at \(x=2\) vanishes.

Solution

The derivative of the given function is obtained as

\[\frac{\mathrm{d}f_a}{\mathrm{d}x} = (1+ax)\mathrm{e}^{ax}\]

so that

\[\left.\frac{\mathrm{d}f_a}{\mathrm{d}x}\right\vert_{x=2} = (1+2a)\mathrm{e}^{2a}.\]

As a consquence, the derivate vanishes provided \(1+2a=0\), i.e. for \(a=-1/2\).

This calculation can be checked by means of Sage: