Assume there is a function $f(x)$ continuous on a closed interval $[\xi,\eta]$. Is it possible to show that there exists a point $\gamma\in[\xi,\eta]$ such that the following is true?
$$ |a f(\xi) + b f(\eta)| \leq |a+b| |f(\gamma)|$$
UPD: $a$ and $b$ are given real numbers.
No, not in general. For instance if $a=1$ and $b=-1$ the right-hand side will always be $0$, but the left-hand side will be positive if $f(\xi)\neq f(\eta)$. Even if you require $a+b\neq 0$ it will still not be true, since you can start with a counterexample where $a+b=0$ and then perturb $b$ by a small amount and the right-hand side will stay close to $0$ (uniformly in $\gamma$ since $f$ is bounded).