Find the integers $n \in \mathbb{N}^{*}$ such that the function $x \mapsto x^{n}$ is convex.
The second derivative of the function $x \mapsto x^{n}$ is $x \mapsto n(n-1) x^{n-2}$, which is positive on $\mathbb{R}$ if and only if $n=1$ or if $n$ is even. Consequently, $x \mapsto x^{n}$ is convex if and only if $n=1$ or if $n$ is even.