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 $\;f(x_1,y_1)<f(x,y)<f(x_2,y_2)\;$?
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.