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For a given $c^*$, suppose that the following system of non-linear equations in $x$ and $y$,

$f(x,y;c)=0\\ g(x,y;c)=0$

possesses a unique solution $(x^*,y^*)$. The equations are such that I do not have $x$ and $y$ as explicit functions of $c$.

Now I have an expression $h(x,y;c)$. I want to prove that $h(x^*,y^*;c^*)>0$

What are some strategies that I can follow? If I just had a function, I could look for other functions to bound it by but since I have these two constraints, I am not sure how to proceed. I would love to get a variety of suggestions to attack this. Thanks.

An example of the kinds of equations I am running into. In this example, $g(x,y;c)$ is the derivative of $f(x,y;c)$ with respect to $x$. The $h(x,y;c)$ are expressions of $\frac{\delta x}{\delta I}$ and $\frac{\delta y}{\delta I}$ where $I$ is some variable that I have suppressed in this question and I got these expressions assuming that Implicit Function Theorem holds.

$f(x,y;c)=12.8428\, -\frac{0.213828 x^{2.97166} \left(-2392.97 (c-0.057)+31.5023 y^{0.15}-0.450832 y\right)}{y^{2.97166}}+\frac{0.213828 y^{1.70499} \left(4170.75 (c-0.057)-47.9187 y^{0.15}+0.32861 y\right)}{x^{1.70499}}-1403.51 c+16.9824 x^{0.15}+0.833333 x$

$g(x,y;c)=0.833333\, -\frac{0.364576 y^{1.70499} \left(4170.75 (c-0.057)-47.9187 y^{0.15}+0.32861 y\right)}{x^{2.70499}}-\frac{0.635424 x^{1.97166} \left(-2392.97 (c-0.057)+31.5023 y^{0.15}-0.450832 y\right)}{y^{2.97166}}+\frac{2.54737}{x^{0.85}}$

Two examples of $h(x,y;c)$

$h_1(x,y;c)=\frac{0.213828 y^{1.70499} \left(0.32861\, -\frac{7.1878}{y^{0.85}}\right)}{x^{1.70499}}+\frac{0.635424 x^{2.97166} \left(-2392.97 (c-0.057)+31.5023 y^{0.15}-0.450832 y\right)}{y^{3.97166}}+\frac{0.364576 y^{0.704995} \left(4170.75 (c-0.057)-47.9187 y^{0.15}+0.32861 y\right)}{x^{1.70499}}-\frac{0.213828 x^{2.97166} \left(\frac{4.72535}{y^{0.85}}-0.450832\right)}{y^{2.97166}}$ $h_2(x,y;c)=\left(\frac{0.364576 y^{1.70499} \left(0.32861\, -\frac{7.1878}{y^{0.85}}\right)}{x^{2.70499}}+\frac{0.213828 y^{1.70499} \left(0.32861\, -\frac{7.1878}{y^{0.85}}\right)}{x^{1.70499}}+\frac{0.635424 x^{2.97166} \left(-2392.97 (c-0.057)+31.5023 y^{0.15}-0.450832 y\right)}{y^{3.97166}}+\frac{0.621599 y^{0.704995} \left(4170.75 (c-0.057)-47.9187 y^{0.15}+0.32861 y\right)}{x^{2.70499}}-\frac{1.88827 x^{1.97166} \left(-2392.97 (c-0.057)+31.5023 y^{0.15}-0.450832 y\right)}{y^{3.97166}}+\frac{0.364576 y^{0.704995} \left(4170.75 (c-0.057)-47.9187 y^{0.15}+0.32861 y\right)}{x^{1.70499}}-\frac{0.213828 x^{2.97166} \left(\frac{4.72535}{y^{0.85}}-0.450832\right)}{y^{2.97166}}+\frac{0.635424 x^{1.97166} \left(\frac{4.72535}{y^{0.85}}-0.450832\right)}{y^{2.97166}}\right)/\left(\left(\frac{0.986175 y^{1.70499} \left(4170.75 (c-0.057)-47.9187 y^{0.15}+0.32861 y\right)}{x^{3.70499}}-\frac{1.25284 x^{0.971661} \left(-2392.97 (c-0.057)+31.5023 y^{0.15}-0.450832 y\right)}{y^{2.97166}}-\frac{2.16526}{x^{1.85}}\right) \left(\frac{0.213828 y^{1.70499} \left(0.32861\, -\frac{7.1878}{y^{0.85}}\right)}{x^{1.70499}}+\frac{0.635424 x^{2.97166} \left(-2392.97 (c-0.057)+31.5023 y^{0.15}-0.450832 y\right)}{y^{3.97166}}+\frac{0.364576 y^{0.704995} \left(4170.75 (c-0.057)-47.9187 y^{0.15}+0.32861 y\right)}{x^{1.70499}}-\frac{0.213828 x^{2.97166} \left(\frac{4.72535}{y^{0.85}}-0.450832\right)}{y^{2.97166}}\right)\right)$

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    My guess is that you could try to bound/approximate $f$ and $g$ and estimate the error in the region of the $x,y$-plane you are interested in (not on the whole plane), with better-shaped and better-behaved functions depending on $c$, then you could try to see how much this changes the intersections by plotting them and modify $h$ consequently by a small change to move the boundary $h=0$ of a lesser quantity. This way (hope it was understandable) you would end up with three functions with better expressions, on which you can probably perform some computations, of algebra or calculus. my$2$cents2012-12-15

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