# Absolute minimum and maximum of multivariable functions

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##### Intros

###### Lessons

**Local Minimum & Maximum of Multivariable Functions Overview:**__Types of Regions__- Open Region
- Closed Region
- Bounded Region
- Examples

__Extreme Value Theorem__- Closed, Bounded region $D$
- Absolute Minimum $f(x_0,y_0)$
- Absolute Maximum $f(x_1,y_1)$

__Steps to Finding Absolute Maximums & Minimums__- Find all critical points inside $D$
- Find all critical points on the boundary $D$
- Find the function values
- Compare, smallest $\to$ absolute min, largest $\to$ absolute max

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##### Examples

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###### Topic Notes

__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__- Closed Region: A region in $\mathbb{R}^2$ that includes its boundary
- Open Region: A region in $\mathbb{R}^2$ that excludes any of the boundary points
- 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 $D$.

- Find all the critical points that are inside the region $D$.
- Find all the critical points on the boundary of region $D$.
- Find the function values for all the critical points
- 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.

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