Consider differentiable functions $f:\mathbb{R} \rightarrow \mathbb{R}$ with the property that for all $a,b \in \mathbb{R}$ we have:
$$f\left ( b \right )-f\left ( a \right )=\left ( b-a \right ){f}'\left ( \frac{a+b}{2} \right )$$
Then which one of the following sentence is true?
- Every such $f$ is a polynomial of degree less than or equal to $2$
- There exists such a function $f$ which is a polynomial of degree bigger than $2$
- There exists such a function $f$ which is not a polynomial
- Every such $f$ satisfies the condition $f\left ( \frac{a+b}{2} \right )\leq \frac{f\left ( a \right )+f\left ( b \right )}{2}$ for all $a,b \in \mathbb{R}$
