Help: How to Proceed

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

01f2(x)dx=[01f(x)dx]2\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

Note by Upamanyu Mukharji
2 years, 3 months ago

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@Calvin Lin @Geoff Pilling @ Nihar Mahajan

Upamanyu Mukharji - 2 years, 3 months ago

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Hint: Discretize the integration. What does that remind you of?

Calvin Lin Staff - 2 years, 3 months ago

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

Upamanyu Mukharji - 2 years, 3 months ago

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@Upamanyu Mukharji @Calvin Lin Sir please help

Upamanyu Mukharji - 2 years, 3 months ago

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@Upamanyu Mukharji FInd all sequences of real numbers such that

nan2=[an]2 n \sum a_n ^2 = \left[ \sum a_n \right]^2

Calvin Lin Staff - 2 years, 2 months ago

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@Calvin Lin OK sir... then?

Upamanyu Mukharji - 2 years, 2 months ago

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@Upamanyu Mukharji

  1. What is the answer? What is the proof?
  2. How does the discretized version relate to your original question?

Calvin Lin Staff - 2 years, 2 months ago

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@Calvin Lin 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?

Upamanyu Mukharji - 2 years, 2 months ago

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@Upamanyu Mukharji Check out cauchy-schwarz inequality.

Calvin Lin Staff - 2 years, 2 months ago

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