Key Idea: The sign of the derivative shows whether a function goes up or down. You use it to find the intervals where a function increases or decreases.
| Sign | Graph |
|---|---|
| f′(x) > 0 | Increasing |
| f′(x) < 0 | Decreasing |
| f′(x) = 0 | Stationary |
Finding the intervals
- Differentiate — Find f′(x).
- Critical values — Set f′(x) = 0 and solve.
- Test — Test the sign of f′ in each region. A sign diagram shows them all at once.
- State — Write the intervals of increase and decrease.
Positive gradient, going up. Negative gradient, going down.
f(x) = 2x³ − 3x² − 12x + 1. Where is f increasing? f′(x) = 6(x − 2)(x + 1) > 0 for x < −1 or x > 2.
Where is f(x) = x² − 6x + 5 decreasing? f′(x) = 2x − 6 < 0, so x < 3.
What can a point with f′(x) = 0 be? A local maximum, local minimum or point of inflection.
A sales model has S′(t) < 0 for t > 4. What does this mean? Sales fall after t = 4.