Understand the problem

Consider differentiable functions $$f:\mathbb{R} \to \Bbb{R}$$ with the property that for all $$a, b \in \Bbb R$$ we have: $$f (b)-f (a) = (b-a)f'( (a + b)/2)$$.

Then which one of the following sentences is true?

1. Every such $$f$$ is a polynomial of degree less than or equal to $$2$$
2. There exists such a function $$f$$ which is a polynomial of degree bigger than $$2$$
3. There exists such a function $$f$$ which is not a polynomial
4. Every such $f$ satisfies the condition $$f(( a + b)/2)\leq (f (a) + f (b))/2$$ for all $$a, b \in \Bbb R$$
Source of the problem
TIFR 2019 GS Part A, Problem 4
Calculus
Moderate

Introduction to Real Analysis by Donald R. Sherbert and Robert G. Bartle

Do you really need a hint? Try it first!

Prove that, $$f$$ will infinitely differentiable here[Use $$f'(x)=\frac{f(x+y)-f(x-y)}{2y}\Rightarrow f”(x)= \frac{f'(x+y)-f'(x-y)}{2y}$$ and so on
$$f(x+y)-f(x-y)=2yf'(x)$$, $$\forall x,y$$ Differentiating w.r.t $$y$$, $$f^{\prime\prime}(x+y)=f^{\prime\prime}(x-y),\forall x,y$$

Hence $$f^{\prime\prime}(x)$$ is constant, so $$f$$ is a polynomial of degree $$2$$

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