Let $f:\mathbb{R}\to\mathbb{R}$ be a concave function. Let $x_0\in \mathbb{R}$ and $\alpha > 1$. Show that for all $x\in \mathbb{R}$,
$f(x_0+x) + f(x_0-x) \geq f(x_0+\alpha x) + f(x_0-\alpha x).$
It seems true on a picture but I don't see how to obtain it from the definition of a concave function.
By concavity, you have $$\begin{aligned}f(\beta x'+(1-\beta)y')&\geq\beta f(x')+(1-\beta)f(y')\\f((1-\beta)x'+\beta y')&\geq(1-\beta)f(x')+\beta f(y')\end{aligned}$$ for any $\beta\in[0,1]$. Now use $x'=x_{0}-\alpha x$, $y'=x_{0}+\alpha x$ and $\beta=\frac{1-\alpha}{-2\alpha}\in(0,1)$ in the sum of the two inequalities to get what you need.