AP Calculus BC / Unit 5: Analytical Applications of Differentiation
5.6 Determining Concavity of Functions over Their Domains
Notes
- $f''>0$: concave up ($f'$ increasing). $f''<0$: concave down ($f'$ decreasing).
- Inflection point: where $f$ changes concavity (so $f''$ changes sign).
Practice
$f(x)=x^3$ has an inflection point at:
$f''=6x$ changes sign at 0.
$f(x)=x^4$ at $x=0$:
$f''=12x^2$ doesn't change sign.
If $f'$ is decreasing on an interval, then $f$ is:
A decreasing slope means concave down.
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