Finding Maximums & Minimums

Critical points: Where f'(x) = 0 or f'(x) is undefined.

First Derivative Test:

f' changes from + to - at c → local maximum

f' changes from - to + at c → local minimum

f' doesn't change sign → neither (just a flat spot)

Second Derivative Test:

If f'(c) = 0 and f''(c) < 0 → local maximum (concave down)

If f'(c) = 0 and f''(c) > 0 → local minimum (concave up)

If f''(c) = 0 → test is inconclusive, use first derivative test

Absolute extrema on closed interval [a,b]:

1. Find all critical points in (a,b)

2. Evaluate f at critical points AND endpoints

3. Largest value = absolute max. Smallest = absolute min.

This is on EVERY AP exam.

Reference:

Wikipedia: Extrema

https://en.wikipedia.org/wiki/Maxima_and_minima