Let $f \in C^{\infty}([0,+\infty])$ such that.
- $f(0) = 0$, $f(x) \geq 0$ for all $x \in (0,+\infty)$, $f(+\infty) = 1$
- $f'(x) > 0$ for all $x \in (0,+\infty)$, $f'(0) = 1$
- $f''(x) < 0$ for all $x \in (0,+\infty)$, $f''(0) = f''(+\infty) = 0$
- There's a unique $x_0 \in (0,+\infty)$ such that $f'''(x) = 0$.
For fixed $h > 0$ consider the integral function $$ G(z) = \int_{z}^{z+h}f(x)dx-\frac{f(z)+f(z+h)}{2}h $$
How can I prove/disprove under such condition that $G(z)$ has unique maximum? My guess is that the unique maximum does exist. And I'm trying to understand the theorems to be applied to prove it, but there are too many hypotesis that really confuse me. The only thing I was able to spot is that $f$ is a convex function.