Let $f(x)=x^{q-1}\ln{(x)}$, with $q\in [0,2]$. Show that $$f(x)\le f(1-x)$$ for all $x\in(0,1/2]$.
I have plotted the function and I know that this is correct. But when I try to show this analytically, I am getting stuck (for instance, I write the whole expression $f(x)-f(1-x)$ and I try to manipulate it or I take the derivative but neither approach brings much).
Note also that the two expressions are almost equal close to $0$ and equal at $x=1/2$ so rough inequalities will not bring much. Equivalently $f(x)-f(1-x)$ is very close to zero when $x$ is close to $0$ or $x$ close to $1/2$.
Any hints?