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Get Started Now- Intro Lesson: a11:31
- Intro Lesson: b3:19
- Intro Lesson: c11:47

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:

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

There are 3 types of critical points:

- 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.
- 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.
- Saddle point: neither a local minimum or local maximum.

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:

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

- Introduction
**Local Minima & Maxima of Multivariable Functions Overview:**a)__Definition of Critical Points__- Critical points for 1 variable
- Critical points for 2 variables
- An example

b)__Types of Critical Points__- Local maximum
- Local minimum
- Saddle point

c)__Classifying Critical Points__- Calculate $D$
- $D\gt0$ and $f_{xx}(x_0,y_0)\gt0 \to$ local minimum
- $D\gt0$ and $f_{xx}(x_0,y_0)\lt0 \to$ local maximum
- $D\lt0 \to$ saddle point
- $D\lt0 \to$ saddle point
- $D=0 \to$ failed to classify
- An example