Can you add quasiconcave functions with scaling to get another quasiconcave function?

Question

Give two quasiconcave functions $f(t),g(t)$, can we always find some $c>0$ so that $f(t)+cg(t)$ is quasiconcave?

I know that $c=1$ doesn't work in general, see Is the sum of quasi concave functions quasi concave But can you find some $c$ so that it does work? What if you assume that $f,g$ are strictly quasiconcave?

This came up in a research problem and would help a lot.

Answer 1

The same counterexample as in the link you gave: $f(x) = x^3$ and $g(x) = 1 - x^2$ are quasiconcave, but $f(x) + c g(x)$ is not quasiconcave for any $c > 0$ (note that $x=0$ is a strict local maximum but not a global maximum).