Section 4.2

Critical Points

Before we can find the peaks and valleys of a landscape, we need to find the flat spots. Critical points are the "candidates" for maximums and minimums.

Defining the Critical Point

A critical point of a function is a number in the domain where either:

Zero Slope

Peaks, Valleys, or Flat Shelves.

Undefined Slope

Sharp Corners (Cusps) or Vertical Tangents.

Fermat's Theorem: If has a local max or min at , then is a critical point.Note: The reverse is not always true! A critical point isn't always a max/min (e.g., at ).

Worked Example: Polynomials

Find Critical Points

Analyze .

Slope Scanner

Increasing (m ≈ 12.00)

Drag to find where the tangent turns blue (Zero Slope).

1. Find Derivative

2. Set to Zero

Factor:

3. Solve

The critical points are:

Level Up Examples

Example A: Trigonometry

Find critical points of on .

1. Differentiate:

2. Solve f'(x) = 0:

3. Unit Circle Values:

Example B: Where derivative fails

Find critical points of .

1. Differentiate:

2. Check Undefined:

anywhere. However, is undefined when (division by zero).

Critical Point:

Graphically, this is a distinct "cusp" or sharp point.