Types of Critical Points for f'(x)

for when f'(x) = 0 or when f'(x) does not exist

Local Maximum (with Horizontal Tangent)

Local Minimum (with Horizontal Tangent)

Point of Inflection (Increasing, Horizontal Tangent)

Point of Inflection (Decreasing, Horizontal Tangent)

Cusp (a local maximum with undefined tangent)

Cusp (a local minimum with undefined tangent)

Point of Inflection (Increasing, Vertical Tangent)

Point of Inflection (Decreasing, Vertical Tangent)