Prove $ax - x\log(x)$ is convex? How do you prove a function like $ax - x\log(x)$ is convex? The definition doesn't seem to work easily due to the non-linearity of the log function.
Any ideas?

A function $f(x) \in C^2(\Omega)$ is convex if its second derivative is non-negative.
$$f(x) = x \log(x) \implies f'(x) = x \cdot \dfrac1x + \log(x) \implies f''(x) = \dfrac1x > 0$$
EDIT
If $f(x) = ax - x\log(x)$, then
$f''(x) = - \dfrac1x$ and hence the function is concave.
