Absolute minimum and maximum of multivariable functions

Notes:

This section is about optimizing a function, such that you find the absolute minimum and maximum on a certain region in $\mathbb{R}^2$.

Types of Regions

1. Closed Region: A region in $\mathbb{R}^2$ that includes its boundary
2. Open Region: A region in $\mathbb{R}^2$ that excludes any of the boundary points
3. Bounded Region: A region in $\mathbb{R}^2$ that is contained in a disk.

Extreme Value Theorem
If a function $f(x,y)$ is continuous in a closed, bounded region $D$ in $\mathbb{R}^2$, then there are 2 points $(x_0,y_0)$ and $(x_1,y_1)$ where $f(x_0,y_0)$ is the absolute minimum and $f(x_1,y_1)$ is the absolute maximum of the function in the region $D$.

1. Find all the critical points that are inside the region $D$.
2. Find all the critical points on the boundary of region $D$.
3. Find the function values for all the critical points
4. Compare all the function values to see which is the smallest, and which is the largest. The smallest is the absolute minimum & and the largest is the absolute maximum.