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(x0,y0)
- Absolute Maximum f(x1,y1)
- 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 → absolute min, largest → absolute max
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Examples
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Topic Notes
Notes:
If a function f(x,y) is continuous in a closed, bounded region D in R2, then there are 2 points (x0,y0) and (x1,y1) where f(x0,y0) is the absolute minimum and f(x1,y1) is the absolute maximum of the function in the region D.
This section is about optimizing a function, such that you find the absolute minimum and maximum on a certain region in R2.
Types of Regions- Closed Region: A region in R2 that includes its boundary
- Open Region: A region in R2 that excludes any of the boundary points
- Bounded Region: A region in R2 that is contained in a disk.
If a function f(x,y) is continuous in a closed, bounded region D in R2, then there are 2 points (x0,y0) and (x1,y1) where f(x0,y0) is the absolute minimum and f(x1,y1) 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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