Finding Extrema of Multivariable Functions

Method for Finding Extrema

In a region

First, you must find possible extrema in the interior. This is true when $f_x = 0$ and $f_y = 0$, or they're both undefined.

Next, you must check the boundary. So, you can either use Lagrange multipliers $f(x,y) = \lambda{}g(x,y)$, or you can just plug it in. Take this question for example $$f(x,y) = x^2 + 4y^2 - x + 2y \text{ , and the boundary is } x^2 + 4y^2 = 1.$$ On the boundary, this becomes $f(x) = (x^2 + 4y^2) - x \pm \sqrt{1 - x^2} = 1 - x \pm \sqrt{1 - x^2}$. Now, you need to find a critical point of that, so you take the derivative. $$f'(x) = 0 = -1 \pm \dfrac{-x}{\sqrt{1- x^2}} \rightarrow x = \pm \frac{\sqrt{2}}{2}$$ So, after doing this, you compile a list of possible absolute maxima on that region. You can just plug in the points and find the largest and smallest values.

No Boundary

Without a boundary, you can only find local maxima, so you can't just plug in all the points and find the largest ones. 

You must find critical points again, when the partials are both zero or undefined, but this time, you must plug in the multivariable second derivative test. Depending on what the values of g and $f_{xx}$ are, it's either a local minimum, a local maximum, or a saddle point.