Section 14.4

Absolute Extrema

Finding the absolute highest and lowest points on a map.

1

Introduction

In 1D calculus, to find the absolute max/min on , we checked critical points AND the endpoints and .

In 3D, our "endpoints" become a boundary curve. We calculate the extrema on the inside (critical points) and on the edge (boundary).

2

Extreme Value Theorem

If is continuous on a closed, bounded set , then attains an absolute maximum and an absolute minimum on .

"Closed" means it includes the boundary (like vs ). "Bounded" means it's finite in size.

3

The Process

Step 1: Inside

Find critical points inside the region.

  • Solve .
  • Keep only points inside .
  • Evaluate at these points.

Step 2: Boundary

Maximize/Minimize on the edge.

  • Parameterize the boundary (e.g. or ).
  • Turn it into a 1D calculus problem.
  • Evaluate at these boundary extrema.

Interactive: Constrained Max/Min

4

Worked Examples

Example 1: Extrema on a Square

Find absolute extrema of on the square .

Step 1: Check Interior

. is on boundary, technically. Value: -1.

Step 2: Check Boundaries

  • Corners: , , , .
  • Edges: Min of is at (-1).

Step 3: Compare

Max: 4. Min: -1.

Example 2: Triangular Region

Maximize on triangle with vertices .

1. Critical Points:

. Value: 0.

2. Boundary:

  • (bottom): .
  • (left): .
  • Hypotenuse (): Substitute into f:

.
.
If . Value .

Absolute Max: 2. Absolute Min: 0.

Example 3: Circular Boundary

Find Absolute Max of on disk .

1. Interior:

. Value: 0.

2. Boundary:

Circle .
Sub into f: for .

Max of is at (Value 2). Min at (Value 1).

Absolute Max: 2 (at ).
Absolute Min: 0 (at ).

5

Practice Quiz

Loading...