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

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

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

#Calculus

Easy Math Editor

This discussion board is a place to discuss our Daily Challenges and the math and science related to those challenges. Explanations are more than just a solution — they should explain the steps and thinking strategies that you used to obtain the solution. Comments should further the discussion of math and science.

When posting on Brilliant:

Use the emojis to react to an explanation, whether you're congratulating a job well done , or just really confused .
Ask specific questions about the challenge or the steps in somebody's explanation. Well-posed questions can add a lot to the discussion, but posting "I don't understand!" doesn't help anyone.
Try to contribute something new to the discussion, whether it is an extension, generalization or other idea related to the challenge.
Stay on topic — we're all here to learn more about math and science, not to hear about your favorite get-rich-quick scheme or current world events.

Markdown	Appears as
`italics` or `_italics_`	italics
`bold` or `__bold__`	bold
- bulleted - list	bulleted list
1. numbered 2. list	numbered list
Note: you must add a full line of space before and after lists for them to show up correctly
paragraph 1 paragraph 2	paragraph 1 paragraph 2
`[example link](https://brilliant.org)`	example link
`> This is a quote`	This is a quote
# I indented these lines # 4 spaces, and now they show # up as a code block. print "hello world"	# I indented these lines # 4 spaces, and now they show # up as a code block. print "hello world"

Math	Appears as
Remember to wrap math in `$` ... `$` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$

Comments

@Calvin Lin @Geoff Pilling @ Nihar Mahajan

Upamanyu Mukharji - 3 years, 11 months ago

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

Calvin Lin Staff - 3 years, 11 months ago

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 - 3 years, 11 months ago

@Upamanyu Mukharji – @Calvin Lin Sir please help

Upamanyu Mukharji - 3 years, 11 months ago

@Upamanyu Mukharji – FInd all sequences of real numbers such that

$n \sum a_n ^2 = \left[ \sum a_n \right]^2$

Calvin Lin Staff - 3 years, 10 months ago

@Calvin Lin – OK sir... then?

Upamanyu Mukharji - 3 years, 10 months ago

@Upamanyu Mukharji –

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

Calvin Lin Staff - 3 years, 10 months ago

@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 - 3 years, 10 months ago

@Upamanyu Mukharji – Check out cauchy-schwarz inequality.

Calvin Lin Staff - 3 years, 10 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$