5.9 Connecting f, f', and f''

What each tells you

The sign of f' gives increasing/decreasing; the sign of f'' gives concavity. A local extremum is where f' = 0 and changes sign; an inflection point is where f'' = 0 and changes sign.

Combining

Read them together: e.g. f' > 0 and f'' < 0 means the graph is increasing and concave down.

Key terms

Sign of f'

Increasing (+) or decreasing (−).

Sign of f''

Concave up (+) or concave down (−).

Local extremum

f' = 0 with a sign change.

Inflection point

f'' = 0 with a sign change.

Combined

e.g. f' > 0, f'' < 0 = increasing, concave down.

Try an AP-style question

f'(2) = 0 and f''(2) = −2. What is true about f at x = 2?

1. f has an inflection point
2. f is increasing
3. f has a relative minimum
4. f has a relative maximum