Increasing, decreasing, and extrema
What you'll learn
Use the sign of f′ to find where a function rises, falls, and reaches its local maxima and minima.
The derivative tells you the slope — and the slope tells you the shape. The sign of f′ alone reveals where a function climbs, where it descends, and where it peaks or bottoms out.
Sign of f′ = direction of f
The rule is simple and exact:
- f′(x) > 0 on an interval → f is increasing there.
- f′(x) < 0 → f is decreasing.
- f′(x) = 0 → the tangent is flat: a critical point, a candidate for a peak or valley.
The first-derivative test
Critical points are only candidates. To classify one, look at how f′ changes sign as you pass through it:
| f′ changes | at the critical point |
|---|---|
| + then − | local maximum |
| − then + | local minimum |
| no sign change | neither (a plateau or a saddle-like flat) |
It matches intuition: a peak is where the curve stops climbing and starts falling, so its slope goes from positive to negative.
Finding them, step by step
For f(x) = ½x³ − 1.5x:
- Differentiate: f′(x) = 1.5x² − 1.5.
- Set to zero: 1.5x² − 1.5 = 0 → x² = 1 → x = ±1 (the critical points).
- Test signs: f′ is positive for x < −1, negative between −1 and 1, positive for x > 1. So x = −1 is a local max (+ to −) and x = 1 is a local min (− to +).
Why this matters
This is how you find the best and worst points of anything a function models — peak power, minimum drag, highest profit — without plotting the whole graph. Set the derivative to zero, then read the sign changes.
Check your understanding
Question 1 of 2
If f′(x) > 0 on an interval, then on that interval f is: