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Applications of Derivatives on polynomials: TIFR 2019 GS Part A, Problem 4

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
Topic
Calculus
Difficulty Level
Moderate
Suggested Book

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

 

Start with hints

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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