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coolnessitself

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## Homework Statement

[tex]\int\limits_{\theta}^{\infty} f(x) g(x) dx = 0 [/tex]

[tex]\theta > 0 [/tex]

[tex]f(x) = ke^{-k(x-\theta)}[/tex]

Show g(x) is identically 0.

## Homework Equations

## The Attempt at a Solution

[tex]f[/tex] is always >= 0 since it behaves exponentially in the region of interest.

From something like https://www.physicsforums.com/showthread.php?t=299145" I could say that IF [tex]f(x)g(x)\ge 0[/tex] in this region, then [tex]g(x)=0[/tex], but I don't know anything about g. I could say that assuming g is positive somewhere, then the integral wouldn't be zero, and that assuming g is negative somewhere, the integral wouldn't be zero, but what about the case where [tex]g(c_1) f(c_1) = k[/tex] and [tex]g(c_2) f(c_2) = -k[/tex]. Then g is positive somewhere and negative somewhere else such that the product cancels out. Wouldn't that allow the integral to be zero?

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