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I have some implicitly defined functions and I need to prove that

$f_1(x)+f_2(x)-x$ has an interior optimum on some closed interval $[a,b]$ where $a>0$

I also know that:

$f_1(x) > 0$ and is monotonically increasing.

$f_1(x) <0$ and is monotonically decreasing.

Since my functions are implicitly defined via a system of non-linear equations, I don't have $f_1(x), f_2(x)$ defined in terms of elementary functions. I am hoping to find some conditions on their derivatives or something to prove the result.

Thanks

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    @coffeemath sweet. That's a great suggestion. I think I should be able to test that.. or at least find bounds at the boundary and find something inside that beats the bounds. Thanks!2012-11-10

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