Absolute minimum and maximum of multivariable functions

Absolute minimum and maximum of multivariable functions

Lessons

Notes:

This section is about optimizing a function, such that you find the absolute minimum and maximum on a certain region in R2\mathbb{R}^2 .

Types of Regions
  1. Closed Region: A region in R2\mathbb{R}^2 that includes its boundary
  2. Open Region: A region in R2\mathbb{R}^2 that excludes any of the boundary points
  3. Bounded Region: A region in R2\mathbb{R}^2 that is contained in a disk.
Extreme Value Theorem
If a function f(x,y)f(x,y) is continuous in a closed, bounded region DD in R2\mathbb{R}^2, then there are 2 points (x0,y0)(x_0,y_0) and (x1,y1)(x_1,y_1) where f(x0,y0)f(x_0,y_0) is the absolute minimum and f(x1,y1)f(x_1,y_1) is the absolute maximum of the function in the region DD.
  1. Find all the critical points that are inside the region DD.
  2. Find all the critical points on the boundary of region DD.
  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.
  • Introduction
    Local Minimum & Maximum of Multivariable Functions Overview:
    a)
    Types of Regions
    • Open Region
    • Closed Region
    • Bounded Region
    • Examples

    b)
    Extreme Value Theorem
    • Closed, Bounded region DD
    • Absolute Minimum f(x0,y0)f(x_0,y_0)
    • Absolute Maximum f(x1,y1)f(x_1,y_1)

    c)
    Steps to Finding Absolute Maximums & Minimums
    • Find all critical points inside DD
    • Find all critical points on the boundary DD
    • Find the function values
    • Compare, smallest \to absolute min, largest \to absolute max