Convex function: Comparing slopes of secant and tangent lines

Question

I have this problem that I feel would be easy to show. However, since I can't show it I am not sure that it is true.

Suppose $f(x)$ continuous on $\mathbb{R}_+$. $f(0)=0$ and $f',f''>0$. Now given $0\leq x_1<x_2<x_3$ and $|x_3-x_2|>|x_2-x_1|$ show that:

$\frac{f(x_2)-f(x_1)}{f'(x_1)(x_2-x_1)}<\frac{f(x_3)-f(x_2)}{f'(x_2)(x_3-x_2)}$

I have tried using the mean value theorem but I get stuck. Anybody that can help me?

Answer 1

I don't think this inequality is true. My calculation show that the inequality fails when $f(x)=x^{2}, x_1=1,x_2=2.1, x_3=4$.