0% Complete
0/59 Steps

# 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

## Application of eigenvalue in degree 3 polynomial: ISI MMA 2018 Question 14

This is a cute and interesting problem based on application of eigen values in 3 degree polynomial .Here we are finding the determinant value .

## Order of rings: TIFR GS 2018 Part B Problem 12

This problem is a cute and simple application on the ring theory in the abstract algebra section. It appeared in TIFR GS 2018.

## Last three digit of the last year: TIFR GS 2018 Part B Problem 9

This problem is a cute and simple application on the number theory in classical algebra portion. It appeared in TIFR GS 2018.

## Group in graphs or graphs in groups ;): TIFR GS 2018 Part A Problem 24

This problem is a cute and simple application on the graphs in groups in the abstract algebra section. It appeared in TIFR GS 2018.

## Problems on quadratic roots: ISI MMA 2018 Question 9

This problem is a cute and simple application on the problems on quadratic roots in classical algebra,. It appeared in TIFR GS 2018.

## Are juniors countable if seniors are?: TIFR GS 2018 Part A Problem 21

This problem is a cute and simple application on the order of a countable groups in the abstract algebra section. It appeared in TIFR GS 2018.

## Coloring problems: ISI MMA 2018 Question 10

This problem is a cute and simple application of the rule of product or multiplication principle in combinatorics,. It appeared in TIFR GS 2018.

## Diagonilazibility in triangular matrix: TIFR GS 2018 Part A Problem 20

This problem is a cute and simple application on the diagonilazibility in triangular matrix in the abstract algebra section. It appeared in TIFR GS 2018.

## Multiplicative group from fields: TIFR GS 2018 Part A Problem 17

This problem is a cute and simple application on the Multiplicative group from fields in the abstract algebra section. It appeared in TIFR GS 2018.

## Group with Quotient : TIFR GS 2018 Part A Problem 16

This problem is a cute and simple application on Group theory in the abstract algebra section. It appeared in TIFR GS 2018.