Show that $f\left(\frac{x_1+x_2}{2}\right)\leq\frac{f(x_1)+f(x_2)}{2}$ Let $f$ be twice differentiable in $(a,b)$ and $f''>0$ in the same interval. If $a<x_1<x_2<b,$ show that $$f\left(\frac{x_1+x_2}{2}\right)\leq\frac{f(x_1)+f(x_2)}{2}.$$
I'm not sure how to start. I know at least that $f'$ is strictly increasing in $(a,b).$ Should I check separately for the cases $x_1\neq x_2$ and $x=x_2$? I'm supposed to use the mean value theorem, but I don't know how. Any insight I need to make in order to start? 

Rewrite this as 
$f(\frac{x_1+x_2}{2}) - f(x_{1}) \leq f(x_2) - f(\frac{x_1+x_2}{2}) $ and then apply the Mean Value Theorem, noting that $f'$ is increasing.
