(a) Define what it means for a function $g: \mathbb{R} \rightarrow \mathbb{R}$ to be convex. Assuming $g^{\prime \prime}$ exists, state an equivalent condition. Let $f(x)=x \log x$, defined on $x>0$. Show that $f(x)$ is convex.

(b) Find the Legendre transform $f^{*}(p)$ of $f(x)=x \log x$. State the domain of $f^{*}(p)$. Without further calculation, explain why $\left(f^{*}\right)^{*}=f$ in this case.