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?

- 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 \(f(( a + b)/2)\leq (f (a) + f (b))/2\) for all \(a, b \in \Bbb R\)

TIFR 2019 GS Part A, Problem 4

Calculus

Moderate

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 \(2\)

The higher mathematics program caters to advanced college and university students. It is useful for I.S.I. M.Math Entrance, GRE Math Subject Test, TIFR Ph.D. Entrance, I.I.T. JAM. The program is problem driven. We work with candidates who have a deep love for mathematics. This program is also useful for adults continuing who wish to rediscover the world of mathematics.