# Is my claim true for extrema of the given two-variable function?

I have this two-variable function $$f(x,y)=g(x,y) \;A + h(x,y) \;B \qquad\qquad (1)$$ where $$-3 and $$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).

Then, can I claim that $$\;f(x_1,y_1)?

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.

• Thanks, how can I check the boundary of the function domain? And which function do you mean, $f$ or $h,g$? – charmin Mar 29 at 14:53
• 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). – user Mar 29 at 14:58