Maxima of increasing function multiplied by its symmetryc

Question

Let $f$ be a function with the following properties:

• $f$ is strictly increasing
• $f(0) = 0$
• $f(1) = 1$

What are the maxima of $g(x) = f(x)f(1 - x)$? More specifically:

• Under which conditions do we have that $x=\frac{1}{2}$ is the only maximum for $g$?
• Can there be some $f$ so that $g(\frac{1}{2})$ is not the absolute maximum of $g$?

Answer 1

There exists a counterexample actually: take $f(1/2)=1/2, f(4/5)=99/100, f(1/5)=1/2-1/10000$ for example. You can construct even continuous and differentiable such functions so I am not sure on what conditions you have to impose on the function to get your desired result.