If $f'' \ge 0$ in interval $[0, 2]$, prove that $$\int_0^2 f(x)dx \ge 2f(1)$$
The question is graphically trivial I think, but not in mathematically.
I wanted to use the fact that there $\exists c$ s.t. $$\int_0^2f(x)dx = 2f(c)$$ and the Jensen's Thm, which is $$f(tx_1+(1-t)x_2)\le tf(x_1)+(1-t)f(x_2)$$
I tried to find the characteristics that $c$ can have, but it was still hard for me to get an idea. Could you please give some key points to the problem? Thanks.