# 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 $$2$$

# Connected Program at Cheenta

#### College Mathematics Program

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.

# Similar Problems

## Sequences & Subsequences : IIT 2018 Problem 10

This problem appeared in IIT JAM 2018 whch pricisely reqiures concepts of sequences and subsequences from mathematical field real analysis

## Cyclic Groups & Subgroups : IIT 2018 Problem 1

This is an application abstract algebra question that appeared in IIT JAM 2018. The concept required is the cyclic groups , subgroups and proper subgroups.

## Acute angles between surfaces: IIT JAM 2018 Qn 6

This is an application analysis question that appeared in IIT JAM 2018. The concept required is the multivarible calculus and vector analysis.

## Finding Tangent plane: IIT JAM 2018 problem 5

What are we learning?Gradient is one of the key concepts of vector calculus. We will use this problem from IIT JAM 2018 will use these ideasUnderstand the problemThe tangent plane to the surface $latex z= \sqrt{x^2+3y^2}$ at (1,1,2) is given by $$x-3y+z=0$$...

## An excursion in Linear Algebra

Did you know Einstein badly needed linear algebra? We will begin from scratch in this open seminar and master useful tools on the way. The open seminar on linear algebra is coming up on 14th November 2019, (8 PM IST).

## Linear Algebra total recall (Open Seminar)

Open Seminar on linear algebra. A review of all major ideas. Even if you have little or no knowledge about Linear Algebra, you may join. Register now.

## 4 questions from Sylow’s theorem: Qn 4

I have prepared some common questions ok application of Sylow’s theorem with higher difficulty level. It is most propably cover all possible combination.

## 4 questions from Sylow’s theorem: Qn 3

I have prepared some common questions ok application of Sylow’s theorem with higher difficulty level. It is most propably cover all possible combination.

## 4 questions from Sylow’s theorem: Qn 2

I have prepared some common questions ok application of Sylow’s theorem with higher difficulty level. It is most propably cover all possible combination.

## 4 questions from Sylow’s theorem: Qn 1

I have prepared some common questions ok application of Sylow’s theorem with higher difficulty level. It is most propably cover all possible combination.