AP Calculus BC / Unit 5: Analytical Applications of Differentiation
5.9 Connecting a Function, Its First Derivative, and Its Second Derivative
Notes
| $f$ | $f'$ | $f''$ |
|---|---|---|
| increasing | positive | — |
| decreasing | negative | — |
| concave up | increasing | positive |
| concave down | decreasing | negative |
| local extremum | changes sign | — |
| inflection point | local extremum | changes sign |
Practice
If $f''<0$ on $(0,3)$, then on $(0,3)$:
$f''$ is the slope of $f'$.
$f'$ is positive and decreasing. Then $f$ is:
Positive $f'$: rising. Decreasing $f'$: concave down.
An inflection point of $f$ matches what on the graph of $f'$?
$f''$ changes sign there, so $f'$ turns around.
Ready for more? Take the Unit 5 test.