f $\in$ C$^2$[a,b], and f has at least three distinct roots in [a,b]. I'm required to show that there's a point x$_0$ in [a,b], such that f(x$_0$) + f''(x$_0$) = 2f'(x$_0$).
I concluded that there are points y$_1$ and y$_2$ such that f'(y$_1$) and f'(y$_2$) are zero. and a z$_1$ such that f''(z$_1$) = 0. I used the taylor formula at these points, I also considered to define a function g(x) = f(x) - f'(x) so that I have to show that g(x$_0$) = g'(x$_0$), but it didn't help. I also tried to find some counter examples, but didn't succeed. What should I do?
Hint: $g(x) = e^{-x}f(x)$ has also three distinct roots, so $g''(x_0) = 0$ for some $x_0 \in (a, b)$.