Is the integral of a non negative function convex

Question

Let's say the $f(x)$ is a nonnegative function on the interval $[0,x_{max}]$.

$g(x) = \int_{0}^x f(t)\ dt$ where $x \leq x_{max}$

Why is g(x) quasi-convex if $f(x) \geq 0$ on the interval $[0,x_{max}]$?

Answer 1

Nonnegative has nothing to do with it. If you restrict to the case where $f\in C^1([0,x_{max}])$ the FTC becomes handy.

$g'(x)=f(x)$ and $g''(x)=f'(x)$.

So then $g$ is convex if...