I have the following function $f:[0,\frac{1}{2}] \to \mathbb{R}$:
$f(p) = p^2(\log(p))^2 - (1-p)^2(\log(1-p))^2 + (1-2p)\log(p)\log(1-p) + (1-2p)\{p\log(p)+(1-p)\log(1-p)\}$
The inequality I need to show is $f(p) \leq 0$I can show that $f(0) = f(1/2) = 0$, and that $f'(0) = -1$, $f'(1/2) = 0$. The graph of $f$ looks like
.
Since its not monotonic/convex/concave I'm stuck. Any leads are welcome!