I have a four-variable function
$$ F(x,y,z,w)=h_1(x,y)\;g_1(z,w)+h_2(x,y)\;g_2(z,w) $$
All the variables are reals. I want to determine those values of $(x,y)$ for which the function reaches its maximum and minimum regardless of changing $(z,w)$. How can I find them?
Hint: Find when you can have a stationary point $p=(\bar x,\bar y,\bar z,\bar w) \in \mathbb R^4$.
$F:\mathbb R^4\to \mathbb R$ such that $F(x,y,z,w)=h_1(x,y)g_1(z,w)+h_2(x,y)g_2(z,w)$, where $h_i,g_i:\mathbb R^2\to\mathbb R$, $i=1,2$. Supposing $h_i,g_i$ are differentiable, you can impose $$\nabla F=(F'_x,F'_y,F'z,F'_w)=(\underbrace{g_1(z,w)\dfrac{\partial}{\partial x}h_1(x,y)+g_2(z,w)\dfrac{\partial}{\partial x}h_2(x,y)}_{=F'_x},\underbrace{g_1(z,w)\dfrac{\partial}{\partial y}h_1(x,y)+g_2(z,w)\dfrac{\partial}{\partial y}h_2(x,y)}_{=F'_y},\underbrace{h_1(x,y)\dfrac{\partial}{\partial z}g_1(z,w)+h_2(x,y)\dfrac{\partial}{\partial z}g_2(z,w)}_{=F'_z},\underbrace{h_1(x,y)\dfrac{\partial}{\partial w}g_1(z,w)+h_2(x,y)\dfrac{\partial}{\partial w}g_2(z,w)}_{F'_w})=(0,0,0,0).$$