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I have a specific problem that Ive generalized here for simplicity.

Let $F(x)=\int^{g(x)}_0h(x,y)dy $

Suppose $F(0)=0$ (with $g(0)>0$)

Now suppose that $h$ is increasing in $y$. Then, it follows that:

$F(x) \leq g(x) h(x,g(x)) $

Does it therefore have to be the case that $h(0,g(0)) = 0$?

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up vote 1 down vote accepted

Not at all. Let $g(x) = 1$ be constant and $h(x,y) = 2y - 1$. Then $h(0,g(0)) = 1$.

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