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I have this two-variable function $$f(x,y)=g(x,y) \;A + h(x,y) \;B \qquad\qquad (1)$$ where $-3<x,y<3$ and $A,B$ are real constants. I want to find the global maximum and minimum of this function (borders of the function).

Computing partial derivatives wrt $x$ and $y$ and solving this system of equation $$ \frac{\partial }{\partial x}f(x,y)=0 , $$ $$ \frac{\partial }{\partial y}f(x,y)=0 , $$ I obtain two solutions $(x_1,y_1)$ and $(x_2,y_2)\;$; they do not depend on $A$ and $B$ in the specific example I have.

Next, substituting these values into $(1)$, I see that $f(x_1,y_1)<f(x_2,y_2)$.

Then, can I claim that $\;f(x_1,y_1)<f(x,y)<f(x_2,y_2)\;$?

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Generally, you cannot claim this. First, the solution of the differential equations are not necessary a local maximum and a local minimum of the function. They can be either of those or a saddle point. Second you should check the boundary of the function domain: the global maximum and/or global minimum can be there.

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  • $\begingroup$ Thanks, how can I check the boundary of the function domain? And which function do you mean, $f$ or $h,g$? $\endgroup$ – charmin Mar 29 at 14:53
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    $\begingroup$ I mean the function $f$. To check the boundary in the case of the rectangular domain just fix one of the variables to the extreme value (e.g. $y=3$). You will obtain a function of $x$ only and can use usual methods to determine its extreme values (they can be either at $x=\pm3$ or at critical points inside the range). $\endgroup$ – user Mar 29 at 14:58

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