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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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    The answer to such a question largely depends on the actual expressions of $f$, $g$ and $h$, so maybe could you specify them?2012-12-15
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    @wisefool Thanks. I just gave one set of actual expressions.2012-12-15
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    @wisefool, be careful what you wish for --- you just might get it.2012-12-15
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    @wisefool I made another edit where I describe the exact relationship between $f,g$ and $h$. The $h$ are basically $\frac{\delta x}{\delta I}$ that I have derived from differentiating $f,g$ with respect to some variable $I$ that I have suppressed.2012-12-15
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    Have you tried to plot them? For going down from c=0.1235 to c=0.1225 (or around there) f(x,y,c)=g(x,y,c)=0 seem to pass from $0$ intersections to $2$ intersections; there should be a value with only one intersection, but for all those values of $c$, the area $30\le y\le x\le70$ where these intersections lie is contained in $h_1<0$...2012-12-15
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    @wisefool I have other restrictions that keep $c \in (0.047,0.08)$, $0 and $101. I can plot these guys at the specific solutions and check the value. I was wondering if there could be an analytical way to approach the problem. I guess I could try bounds or something.2012-12-15
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    well, this gets nastier and nastier ... I was speaking of plotting the graphs because this could tell you how the functions behave and maybe how to approximate or bound them .. for example, it seems that $g$ is more or less linear, while $f$ is shaped like an oval or a series of ovals. Moreover, as curves in $(x,y)$-plane, $f=0$ and $g=0$ seem to change quite rapidly in $c$, while $h=0$ (which is the boundary of the domain you are interested in) seems to be quite unaffected by small changes of $c$.2012-12-15
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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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