This is a question from definite integrals which I am having doubts about. It is given that f(x) is a continuous function defined in the domain of [0,1] such that

\[\int_{0}^{1} f^2(x)dx=[\int_{0}^{1}f(x)dx]^2\]

We then have to find the range of f(x)

I tried applying Leibniz Differentiation on both sides.. but I have a feeling that won't work as the integrals may NOT be equal for any arbitrary value of x between 0 and 1(both inclusive) except for 1. Could anyone help? #Calculus #KVPY

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TopNewest@Calvin Lin @Geoff Pilling @ Nihar Mahajan – Upamanyu Mukharji · 1 month ago

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Hint:Discretize the integration. What does that remind you of? – Calvin Lin Staff · 1 month agoLog in to reply

– Upamanyu Mukharji · 1 month ago

Sir do you mean express it as a infinite series with infinitesimal common difference(sigma)?... I don't know how to do that...Log in to reply

@Calvin Lin Sir please help – Upamanyu Mukharji · 1 month ago

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\[ n \sum a_n ^2 = \left[ \sum a_n \right]^2 \] – Calvin Lin Staff · 4 weeks, 1 day ago

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– Upamanyu Mukharji · 3 weeks, 6 days ago

OK sir... then?Log in to reply

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– Upamanyu Mukharji · 3 weeks, 5 days ago

Sorry if I'm being extremely dense.... my brain has been fried by such a simple problem! Could you please show me the answer and proof?Log in to reply

cauchy-schwarz inequality. – Calvin Lin Staff · 3 weeks, 5 days ago

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