$f:\mathbb{R} \to \mathbb{R}$ we have $f(b)-f(a)=(b-a)f'(\frac{a+b}{2})$ such function is polynomial of degree less than or equal to two.

Question

Consider differential function $f:\mathbb{R} \to \mathbb{R}$ with the property that for all $a,b \in \mathbb{R}$ we have $$f(b)-f(a)=(b-a)f'(\frac{a+b}{2})$$

Then show that every such function is polynomial of degree less than or equal to two.

i was thinking of applying Roll theorem. but got nowhere.please provide a hint

Thanks in advanced