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?

The answer is quite easy if you use derivatives, so here's one which does not use them. Note that for a continuous function $f:P\to\mathbb{R}$, it is sufficient that it satisfies the inequality:
$$f\left(\frac{x}{2}+\frac{y}{2}\right)\geqslant \frac{f(x)}{2}+\frac{f(y)}{2}$$
for all $x,y\in{P}$ to be concave.
By the AM-GM inequality, we have $\frac{x+y}{2}>\sqrt{xy}$ for $x\ne{y}$. Thus, for your function, we have:
$$\frac{ax+ay}{2}-\ln\frac{x+y}{2}>\frac{ax+ay}{2}-\ln\sqrt{xy}=\frac{ax+ay}{2}-\frac{1}{2}\ln{xy}=\left(\frac{ax-\ln{x}}{2}\right)+\left(\frac{ay-\ln{y}}{2}\right)$$
EDIT: In fact, since in our case, the inequality is strict, your function is strictly concave.
