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?

Differentiating twice, we get
$$\frac{d^2}{dx^2}x\log(x) = \frac{1}{x}$$
The second derivative is strictly positive everywhere the function is defined, so $x\log(x)$ is strictly convex.
This implies that $ax - x\log(x)$ is strictly concave.
