Mean Value Theorem and inequality proving without given function

Question

I have a function $f:\mathbb{R}\rightarrow\mathbb{R}$ which can derives twice and I have $f(1)=f(3)=0$ and $f(2)>0$. Using the Mean Value Theorem I have already prove that exists $x_1,x_2\in(1,3)$ with $x_1<x_2$ for which is true that $f'(x_1)+f'(x_2)=0$. Now I want to prove that exists $x_3\in(1,3)$ for which is true that $f''(x_3)<0$. I am thing of using Rolle's Theorem but I don't know how. Any ideas?

Answer 1

BY Lagrange theorem we have $a\in (1,2)$ such that $$f'(a) ={f(2)-f(1)\over 2-1}>0$$ and we have $b\in (2,3)$ such that $$f'(b) = {f(3)-f(2)\over 2-1}<0$$

So we have, again by Lagrange $c\in (a,b)$ such that $$f''(c) = {f'(b)-f'(a)\over b-a}<0$$