Section 14.3

Relative Extrema

Hills, Valleys, and Saddle Points.

Introduction

Just like in Calculus I, we find peaks and valleys by looking for places where the slope is zero.

In 3D, a "peak" (Relative Max) means the surface curves down in all directions. A "valley" (Relative Min) means it curves up. But there's a third option: a Saddle Point.

Critical Points

Definition

A point is a Critical Point if:

This means BOTH and .

Interactive: Types of Critical Points

Second Derivative Test

To classify a critical point, we calculate the Discriminant :

Relative Max

Relative Min

Saddle Point

Concavity conflicts

If , the test is inconclusive.

Worked Examples

Example 1: Finding Critical Points

Find and classify critical points of .

1. Find Partials:

2. Set to 0:

Critical Point:

3. Classify:

. Since : Relative Minimum.

Example 2: Saddle Point

Classify extrema for .

1. Critical Points:

Point: .

2. Second Derivative Test:

Result: Saddle Point at .

Example 3: Multiple Critical Points

Let . Find all local Extrema.