Is maximum of increasing concave functions quasi-concave

Question

It seems to me that the maximum of increasing concave functions is at least quasi-concave, but I have difficulty to prove it. Anyone can help? Or give a counter-example?

Answer 1

You do not need the hypothesis of concavity. Monotone functions are quasi-concave, the maximum of monotone functions is monotone as soon as they are all decreasing or increasing.

Let $f$ and $g$ two increasing function on $I \subset \mathbb{R}$. Then $\forall x,y$ s.t. $x < y$, we have $f(x) \le f(y)$ and $g(x) \le g(y)$. Thus :

$$\max(f(x),g(x))\le \max(f(y),g(y)).$$

That means that the function $x \mapsto \max(f(x),g(x))$ is also increasing, hence quasi-concave by monotonicity.