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.