Local minima and maxima of multivariable functions

Notes:

Definition of Critical Points

Recall for 1 variable functions, a critical point occurs at a point $x_0$ if $f'(x_0)=0$ or $f'(x_0)$ does not exist. The concept is the same for 2 variable functions, except we must modify a few things.

The point $(x_0, y_0)$ is a critical point of $f(x,y)$ if one of the following is true:

1. $f_x(x_0,y_0)=0$, $f_y(x_0,y_0)=0$
2. $f_x(x_0,y_0)$ and/or $f_y(x_0,y_0)$ does not exist

Types of Critical Points

There are 3 types of critical points:

1. Local Maximum: occurs when $f(x_0,y_0) \geq f(x,y)$ for all points $(x,y)$ that is around $(x_0, y_0)$. In other words, it's the biggest value of the function around it's region.
2. Local Minimum: occurs when $f(x_0,y_0) \leq f(x,y)$ for all points $(x,y)$ that is around $(x_0, y_0)$. In other words, it's the smallest value of the function around it's region.
3. Saddle point: neither a local minimum or local maximum.

Classifying Critical Points
Suppose that $(x_0,y_0)$ is a critical point of $f(x,y)$. To see whether it's a local maximum, or local minimum, or saddle point, we compute the following:

$D=f_{xx}(x_0,y_0)\cdot f_{yy}(x_0,y_0)-[f_{xy} (x_0,y_0)]^2$

If:

1. $D\gt0$ and $f_{xx} (x_0,y_0)\gt0$, then it is a local minimum
2. $D\gt0$ and $f_{xx}(x_0,y_0)\lt0$, then it is a local maximum
3. $D\lt0$, then it is a saddle point
4. $D=0$, then it could be any of the 3 types. Need to use other techniques to classify it.