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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.
easy
Suggested Book
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 $$f(x)=x^3$$.

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