Understand the problem

True or False? If \( f: \Bbb R \to \Bbb R \) is differentiable and bijective then \(f^{-1} \) is also differentiable.
Source of the problem
TIFR GS 2017 Entrance Examination Paper
Topic
Real Analysis
Difficulty Level
Easy
Suggested Book
Introduction to Real Analysis, Fourth Edition, English, Paperback, by Robert G. Bartle and Donald R. Sherbert. Introduction to Real Analysis, Seventh Edition, English, Paperback, by S. K. Mapa.

Start with hints

Do you really need a hint? Try it first!

The given statement is a general statement. So, if I can find a function \(f(x) \) which is differentiable and bijective on \( \Bbb R\) but whose inverse function \( f^{-1}\) is not differentiable then I would be able to show that the statement is not generally true. Now, can you find such a function?
Let us consider the function \(f(x) = x^3 \) . Can you now show that the function is contradicting the statement given to us?
The function \( f(x)= x^3 \) is a bijective and differentiable function. It’s inverse is \( x^{\frac{1}{3}} \) which, you can see, is not differentiale at \( x=0\). Hence the statement is false.

Watch the video (Coming Soon)

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

Isomorphism in b/w infinite dim vector sp: TIFR GS 2019, Part B Problem 10

It is a question on isomomorphisms b/w inf dim vector spaces. It was asked in TIFR 2019 GS admission paper. It is a true false question.

Matrix to real line: TIFR GS 2019, Part B Problem 8

It is a lie algebra question on connections b/w matrices and real space. It was asked in TIFR 2019 GS admission paper. It is a true false question.

Homomorphism to Continuous function: TIFR GS 2019, Part B Problem 9

It is a lie algebra question on homomorphisms b/w real ring and ring of continuous function. It was asked in TIFR 2019 GS admission paper. It is a true false question.

Similar matrices: TIFR GS 2019, Part B Problem 7

It is a linear algebra question on similar matrices. It was asked in TIFR 2019 GS admission paper. It is a true false question.

Average Determinant: TIFR GS 2017 Part A Problem 8.

This question has appeared in TIFR GS 2017 Entrance Examination and is based on Linear Algebra.

Spanning set of a matrix space: TIFR GS 2019, Part B Problem 6

It is a linear algebra question on matrices basically on spanning set of a matrix space. It was asked in TIFR 2019 GS admission paper.

ABC of rank: TIFR GS 2019, Part B Problem 5

It is a linear algebra question on matrices. It was asked in TIFR 2019 GS admission paper. It is a true false question.

Continuous map on countable space: TIFR GS 2019, Part B Problem 3

It is a topology question on real plane. It was asked in TIFR 2019 GS admission paper. It is true-false type question.

Invertible Matrix implies identity?: TIFR GS 2019, Part B Problem 4

It is a linear algebra question on matrices. It was asked in TIFR 2019 GS admission paper.

Invertible Matrix: TIFR GS 2019, Part B Problem 2

It is a linear algebra question on matrices. It was asked in TIFR 2019 GS admission paper.