I'm trying to solve a question concerning about average value of function. A problem is
If $f$ is concave upward on $[a,b]$, then $\frac{1}{b-a}\int_a^bf(x)dx > f((a+b)/2)$
I understand why it is true using diagram but I can't prove it. I'm trying to use the fact f is concave upward so $f'$ is increasing but I couldn't go any further. Thank you in advance!