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Continuous Functions: TIFR GS 2017 Part A Problem 1.

Understand the problem

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 \( f(x)=x^3 \forall x \in \Bbb R \).

Source of the problem
TIFR GS 2017 Part A problem 1.
Topic
Continuous functions and Riemann Integrability.
Difficulty Level
easy
Suggested Book
S.K. Mapa, Bertle and Sherbert

Start with hints

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 \( f(x)=x^3 \).

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