Calculus / Unit 6: Applications of Derivatives II
6.1 Increasing, Decreasing and Critical Numbers
Notes
The sign of $f'$ tells whether $f$ rises or falls.
- $f' > 0$ on an interval: $f$ is increasing. $f' < 0$: $f$ is decreasing.
- A critical number is an $x$ in the domain where $f'(x) = 0$ or $f'(x)$ is undefined.
- To find intervals: find critical numbers, then test the sign of $f'$ in each interval between them.
- Example: $f(x) = x^3 - 12x$ has $f'(x) = 3(x - 2)(x + 2)$. It is positive for $x < -2$ and $x > 2$, and negative between.
Practice
Find the critical numbers of $f(x) = x^3 - 3x$.
$f'(x) = 3x^2 - 3 = 0$ gives $x^2 = 1$.
On what interval is $f(x) = x^2 - 4x$ increasing?
$f'(x) = 2x - 4 > 0$ when $x > 2$.
For $f(x) = x^3 - 12x$, on which set is $f$ increasing?
$f'(x) = 3(x^2 - 4)$ is positive when $|x| > 2$.
Ready for more? Take the Unit 6 test.