Is f Uniformly 0 if Integral of f^2 on [a,b] equals 0?

[krcmd1]

given f continuous on [a,b] with a not equal to b,
does
\int{a_b}(f^2) = 0 [\tex]<br /> <br /> imply that f must be uniformly 0?<br /> <br /> (still not able to get Latex right, either. sorry)<br /> <br /> thanks<br /> <br /> KRC

 

[Evales]

I don't know why they used 'f' since 'f' is generally used to be a function not describe them. If we change the f to x it makes this question easier.

I don't know what it is saying when it asks is x is uniformly zero.
If you find the definite integral of x^2 and make it equal to zero you get:
a^3 = b^3

Since a equals b there will never be any area under the curve. I'm sorry I don't know where to progress from here.

 

[d_leet]

I don't know why they used 'f' since 'f' is generally used to be a function not describe them. If we change the f to x it makes this question easier.

I don't know what it is saying when it asks is x is uniformly zero.
If you find the definite integral of x^2 and make it equal to zero you get:
a^3 = b^3

Since a equals b there will never be any area under the curve. I'm sorry I don't know where to progress from here.

Umm... What are you talking about? It seems fairly evident that f is representing a function.

To krcmd1: What happens if f is not everywhere 0 on [a,b]?

 

[HallsofIvy]

If f is continuous then f² is both continuous and non-negative. Suppose there were some point, say x₀ at which f(x_{0[/sup]) is not 0. By continuity, you can find some small neighborhood of x0 in which f² is positive. The integral of f² over that neighborhood is positive. Since there are no negative values of f² to cancel that, the integral of f² [a, b] is larger than or equal to that positive number: it can't be 0.}

 

[krcmd1]

Thank you Halls of Ivy. That is just how I reasoned it. Don't have enough experience to be confident in my reasoning yet.

Ken