Cue Column:

• What makes a critical point?
• How to classify?
• Can a critical point not be an extremum?

Notes Column: A critical point of f is x = c where either f'(c) = 0 or f'(c) does not exist, with f(c) defined. Critical points are CANDIDATES for extrema — not all are extrema.

First Derivative Test: At critical point c, examine the sign of f' just before and just after c:

• f' changes from + to -: LOCAL MAXIMUM at c
• f' changes from - to +: LOCAL MINIMUM at c
• f' does not change sign (+ to + or - to -): NEITHER (inflection point)

Example: f(x) = $x^4$. f'(x) = 4$x^3$. f'(0) = 0, critical point. f' < 0 for x < 0, f' > 0 for x > 0. Sign change: - to +. So x = 0 is a LOCAL MINIMUM.

Example: f(x) = $x^3$. f'(x) = 3$x^2$. f'(0) = 0, critical point. f' > 0 for x < 0 (except at 0) and f' > 0 for x > 0. No sign change. x = 0 is NOT an extremum; it's an inflection point.

Summary: The first derivative test is always conclusive (unlike the second derivative test). Always make a sign chart.