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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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If you can evaluate the $f_i(x)$ at $a,b$, all you would need is to find one point $c$ for which the left side exceeds its value at endpoints, and from that coulde conclude the existence of an interior max. Similar for an interior min. –  coffeemath Nov 10 '12 at 2:24
    
@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! –  Amatya Nov 10 '12 at 3:29
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