Calculus Proof

[quick sparrow]

I tried using the Weighted Mean Value Theorem but I'm not able to prove the single signed property of g(x). Are there any alternate proofs the the question?

[left acorn]

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[wide tendon]

Well if h is a positive continuous function on [a,b] such that it’s integral is 0 on [a,b] what can you say of h ?

[quick sparrow]

It's a weighted function?

[wide tendon]

It’s 0 on [a,b], any positive continuous function on [a,b] which has 0 integral is 0

[quick sparrow]

Oh right yea

[quick sparrow]

But f(x) need not be a positive continuous function alone right?

It could be intersecting the x-axis at some point in the interval and the integral could be 0 too

[wide tendon]

If it’s positive continuous and has 0 integral then it is 0

[quick sparrow]

That is true. But it hasn't been mentioned anywhere that f(x) is positive continuous.

[wide tendon]

Not f

you know that the integral of f(x)g(x) is 0 for all g

[quick sparrow]

Yes

[wide tendon]

what about when ||g=f||

[quick sparrow]

Then integral of f(x)^2 would be 0

[wide tendon]

Yes

[wide tendon]

.

[quick sparrow]

If there exists a point in [a, b] where f(x) is not 0, then f(x)^2 would be greater than 0 at that point.

Which means f(x) has to be 0 at any point in the interval?

[wide tendon]

well yeah f(x)^2=0 => f(x)=0

[quick sparrow]

But what if g is not equal to f?

[wide tendon]

You know that it’s true for any continuous g

That includes f

[quick sparrow]

We're taking a specific case to prove it and generalizing it to the entire interval.

Is there a generic solution to it?

[wide tendon]

Well there is nothing else to prove, you know that the integral of f(x)g(x) is 0 for any continuous function g, if it’s true for any function g then it’s true for g=f so you have that the integral of f(x)^2 is 0, with what I said before this implies that f(x)^2=0 for all x ie f(x)=0 for all x

[quick sparrow]

Got it then

Thanks @wide tendon

[wide tendon]

Np

[quick sparrow]

If it is alright with you, could I contact you in private for a couple more problems of a similar structure?

I'm having a bit of a trouble wrapping my head around those

[wide tendon]

I’m sorry but I’m kinda busy because of uni, but you can make help posts, other people may help

And I’ll comment if I have time

[quick sparrow]

Alright then. Thanks again for the help.

+close

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