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I have the following quantity:

[itex]f = \int_{\mu}^{\infty} (1-F(x))a(x)dx[/itex]

I want to claim that by increasing the following quantity:

[itex]g = \int_{\mu}^{\infty} a(x)f(x)dx[/itex]

then [itex] f [/itex] can only increase. Can I differentiate [itex] f [/itex] with respect

to [itex] g [/itex]? Is the following correct?

[itex] \frac{\partial (\int_{\mu}^{\infty} (1-F(x))a(x)dx)}{\partial (\int_{\mu}^{\infty} a(x)f(x)dx)}[/itex]

[itex] = \frac{\frac{\partial (\int_{\mu}^{\infty} (1-F(x))a(x)dx)}{\partial x}}{\frac{\partial (\int_{\mu}^{\infty} a(x)f(x)dx)}{\partial x}}[/itex]

[itex] = \frac{(1-F(x))a(x)}{f(x)a(x)} [/itex]

[itex] = \frac{1-F(x)}{f(x)} [/itex]

I already know that [itex] a(x) > 0 [/itex] for [itex] x>\mu[/itex] and that [itex] \frac{1-F(x)}{f(x)} [/itex] is positive. Therefore [itex] f [/itex] increases when [itex] g [/itex] is increased. Does this resolve the problem?

Thank you for taking the time to read and answer!