True or False? let \( f : [0,1] \to \Bbb R \) be a continuous function such that \( f(x) \geq x^3\) \(\forall x \in [0,1] \) with \(\int_0^1 f(x)= 1/4 \). Then \( f(x)=x^3 \forall x \in \Bbb R \).

TIFR GS 2017 Part A problem 1.

Continuous functions and Riemann Integrability.

easy

S.K. Mapa, Bertle and Sherbert

Do you really need a hint? Try it first!

We know that continuous functions are integrable, so, how can you use this fact to solve this question?

Since \( f(x) \) and \( x^3 \) are continuous functions so the given inequality can be intregrated over the interval \( [0,1] \).

After integrating the inequality compare the integral value of \( f(x) \) with the given integral value. What can you infer about it?

We can see that \( \int_0^1 f(x) dx \geq 1/4 \) but given value is exactly 1/4. Hence we can conclude that \( f(x)=x^3 \).

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.