Few polynomials satisfy this

#Calculus#Polynomials

Problem Statement

Consider differentiable functions $f: \mathbb{R} \rightarrow \mathbb{R} $ with the property that for all $a,b \in \mathbb{R} $

$$ f(b) - f(a) =(b-a) f'\left(\frac{a+b}{2} \right), $$

then prove that prove that $f$ must be a polynomial of degree $2$ or fewer.

Since $f$ is differentiable once, using the equality $f$ is differentiable twice. Therefore, $f$ is infinitely differentiable. Say $a+b=2x$ and $b-a=2y$, the given equation transforms into:

$$ f(x+y) - f(x-y) = 2y f'(x) $$

Differentiating with respect to $y$, we get

$$ f'(x+y) + f'(x-y) = 2f'(x) $$

Differentiating again with respect to $y$, we get

$$ f''(x+y) - f''(x-y) = 0, \ \text{for all} \ x,y \in \mathbb{R}. $$

$$ \implies f''(x+y) = f''(x-y) , \ \text{for all} \ x,y \in \mathbb{R}. $$

Since $x$ and $y$ can be chosen independently, therefore $x+y$ and $x-y$ are linearly independent and can take any value.

$$ \implies f''(t) = 0, \ \text{for all} \ t \in \mathbb{R}. $$

Therefore, $f$ is a polynomial whose degree is $2$ or fewer.

Bibliography & References

[1]
Calculus