Let $f$ be a twice-differentiable function from R to R. Show that if whenever $f'(c)=\frac{f(b)-f(a)}{b-a}$ then $c=\frac{a+b}{2}$, it must be true that $f$ is a quadratic polynomial.
This is a question on my homework, and I'm a bit stuck. It's obvious that quadratic polynomials are twice-differentiable. I'm thinking the Mean Value Theorem might be involved, but not sure.
Can anyone give me an tips on how to do this?
Thanks.