5.9 Connecting a Function, Its First Derivative, and Its Second Derivative

Reading three graphs as one story

The signs of f' and f'' describe the shape of f. If f' > 0, f is increasing; if f' < 0, f is decreasing. If f'' > 0, f is concave up (f' is increasing); if f'' < 0, f is concave down (f' is decreasing). Combining both tells you increasing/decreasing AND the curvature at the same time.

Matching critical points to behavior

Where f' = 0 and changes sign, f has a local max or min. Where f'' = 0 and changes sign, f has an inflection point (concavity changes). A common exam skill: given the graph of f' (not f!), determine where f is increasing, decreasing, concave up/down, and where extrema or inflection points occur, since f'' is just the slope of f'.

Four sign combinations

f' > 0, f'' > 0: increasing, concave up (speeding up). f' > 0, f'' < 0: increasing, concave down (slowing down). f' < 0, f'' > 0: decreasing, concave up (slowing down). f' < 0, f'' < 0: decreasing, concave down (speeding up). Sketching a small curve for each case helps visualize consistent behavior across f, f', and f''.

Key terms

f' > 0

f is increasing on that interval.

f'' > 0

f is concave up; equivalently, f' is increasing.

Inflection point

A point where f'' changes sign, meaning concavity of f changes.

Local extremum condition

Occurs where f' = 0 (or undefined) and f' changes sign.

Given graph of f', how to find f'' behavior

f'' is the slope of f'; where f' is increasing, f'' > 0 (concave up); where f' is decreasing, f'' < 0 (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 a relative minimum
2. f is increasing
3. f has an inflection point
4. f has a relative maximum