Proving f(x)=0: Int. 0-1 of f(x)x^n

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In summary, the problem states that for a continuous function f on the interval [0,1], if the integral of f(x) times x^n is equal to 0 for all non-negative integers n, then f(x) must be equal to 0. The hint provided is to prove that the integral of f^2 is also equal to 0, which would imply that f(x) must be identically 0. The solution involves using the Stone-Weierstrass theorem and Taylor series to show that the integral of f^2 can be broken down into a series of integrals of the form f(x) * x^n, which are all equal to 0 by assumption. This ultimately proves that f(x) must
  • #1
e(ho0n3
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Homework Statement
Suppose f: [0,1] --> R is continuous and

[tex]\int_0^1 f(x) x^n \, dx = 0[/tex]

for all n = 0, 1, ... Prove that f(x) = 0.

The attempt at a solution
There's a hint that says: Prove that

[tex]\int_0^1 (f(x))^2 \, dx = 0.[/tex]

I don't know how to prove this hint and I don't know how that would help in determining what f is. Any tips?
 
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  • #2
HAH! This is cosmic. Look at this thread.
https://www.physicsforums.com/showthread.php?p=2064400#post2064400
I think the course I was outlining for Hitman2-2 will work perfectly for you, and even more simply since you have the condition for all n>=0. Use Stone-Weierstrass. You have integral f(x)=0 since it's true for n=0. Hitman2-2 only gave me n an even natural number. Is zero an even natural number? I would have said, no. And that's the roadblock for that thread.
 
  • #3
e(ho0n3 said:
Homework Statement
Suppose f: [0,1] --> R is continuous and

[tex]\int_0^1 f(x) x^n \, dx = 0[/tex]

for all n = 0, 1, ... Prove that f(x) = 0.

The attempt at a solution
There's a hint that says: Prove that

[tex]\int_0^1 (f(x))^2 \, dx = 0.[/tex]

I don't know how to prove this hint and I don't know how that would help in determining what f is. Any tips?

Are you SURE you don't know why the integral of f^2 equals zero wouldn't solve the problem?
 
  • #4
The integral of f^2 can only be zero if f is identically zero. If f(x) were greater than zero at some point in the interval, there would be a positive contribution to the integral from that point with no negative contribution to cancel it out (since the square of a real function is non-negative). Think about it.

I don't know how to prove the assertion. I will have to think about it.
 
  • #5
That thread was really helpful. Now I know why the hint is true. However...

I don't know why the integral of f^2 equals zero wouldn't solve the problem. You asked Hitman2-2 the same question as well, but he/she didn't respond.
 
  • #6
Brian_C just answered why the integral of f^2 equal zero solves the problem.
 
  • #7
I have an idea how to prove that the integral of f^2 is equal to zero. Try expanding one of the f(x)'s in the integral of f^2 as a taylor series in x. You should end up with an infinite series of integrals of the form f(x) * x^n, which should all vanish (by assumption), thus proving that the integral of f^2 vanishes. The only problem is, I don't know how to prove that a Taylor series will converge on some interval for this particular function.
 
  • #8
The taylor series doesn't have to converge. The function is only continuous, it doesn't have to be differentiable, much less analytic. Check Hitman2-2's thread. You have to use the Weierstrass approximation theorem.
 
Last edited:

Related to Proving f(x)=0: Int. 0-1 of f(x)x^n

1. What does "Proving f(x)=0" mean?

Proving f(x)=0 means demonstrating that the function f(x) returns a value of 0 for all values of x. In other words, the graph of the function will intersect the x-axis at the point (0,0).

2. What does "Int. 0-1" refer to in this context?

"Int. 0-1" refers to the interval from 0 to 1 on the x-axis. This means that the function will be evaluated and proved to be equal to 0 within this specific range of values for x.

3. What is the significance of the x^n term in the statement?

The x^n term represents an exponential function with a variable exponent n. This means that the function may have a changing rate of increase or decrease, which adds complexity to the proof of f(x)=0.

4. How is this type of proof useful in science?

This type of proof is useful in science because it allows researchers to validate their findings and theories. By proving that a function is equal to 0 within a specific interval, scientists can support their hypotheses and make more accurate predictions about the behavior of systems and phenomena.

5. What are some possible strategies for proving f(x)=0 for this type of function?

Some possible strategies for proving f(x)=0 in this context include using mathematical induction, substitution, and manipulation of algebraic equations. It may also be helpful to use graphical representations to visualize the function and its behavior within the given interval.

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