The function $f(x)$ is strictly increasing, finite, positive and twice continuously differentiable on the compact interval $[0,K]$, and $f(0)=0$.
I'm trying to either find a counterexample to, or a proof of the fact that, for such $f(x)$, the function:
$$g(x) = x f(K-x)$$
is quasiconcave. A few tools/observations I've made along the way:
- If two functions on the same domain are positive, quasiconcave and co-monotone, then their product is quasiconcave
- If two functions are co-monotone and quasiconcave, then their sum is quasiconcave.
- $g(x)$ always has precisely two zeros on $[0,K]$: one at 0 and the other at $K$, and is strictly positive on the interior.
I've tried both the usual proof using quasiconcavity definitions as well as plugged in a large variety of test functions for $f(x)$, and all give me a quasiconcave $g(x)$. If someone can give me a counter-example or any advice on proving a positive result, it would be great!