(a) Define what it means for a function $f: \mathbb{R}^{n} \rightarrow \mathbb{R}$ to be convex.

(b) Define the Legendre transform $f^{*}(p)$ of a convex function $f(x)$, where $x \in \mathbb{R}$. Show that $f^{*}(p)$ is a convex function.

(c) Find the Legendre transform $f^{*}(p)$ of the function $f(x)=e^{x}$, and the domain of $f^{*}$.