Section 4.3

Minimum & Maximum Values

What is the highest point of a function? It depends on your perspective. We distinguish between the highest point in a neighborhood (Local) and the highest point in the universe (Absolute).

The Landscape

Two Types of Extremes

Local (Relative) Extremum

Highest/lowest point in its immediate neighborhood. "King of the Hill".

Occurs at Critical Points.

Absolute (Global) Extremum

Highest/lowest point in the entire domain. "King of the World".

Interactive Demo

The Hill Climber

Zoom Level:

Zoom in to see how a "Local Max" looks like the highest peak ("Absolute") when you ignore the rest of the world.

Worked Example: Parabola

Find Local Extrema

Find the local extrema of .

1. Find Derivative

2. Critical Points

3. Verify & Evaluate

Since parabola opens down, it's a Max.

Answer: Local Maximum of 4 at

Level Up Examples

Example A: Rational Function

Find local extrema of . (Common curve "The Serpent")

1. Quotient Rule:

2. Critical Points (Set Num = 0):

(Denominator never zero, so no undefined points).

3. Classify:

Test signs of f'(x):

is a Local Max (Slope + to -). Value: .
is a Local Min (Slope - to +). Value: .

Example B: Piecewise

Let . Find extrema.

1. Analyze Derivative:

for

is undefined (Corner).

2. Critical Point:

is a critical point.

Since and everywhere else, is the Absolute Minimum.