Application of Derivatives
Increasing and decreasing functions
Find the intervals where a function increases or decreases.
Read a graph from left to right. Where it goes uphill, the function is increasing: bigger x gives bigger f(x). Where it goes downhill, it is decreasing.
The derivative is the slope, so it tells you which way the graph goes:
- f′(x) > 0 on an interval: increasing there.
- f′(x) < 0 on an interval: decreasing there.
- f′(x) = 0 everywhere on an interval: constant.
Method to find the intervals:
- Find f′(x) and factorise it.
- Solve f′(x) = 0. These points split the number line into pieces.
- Pick one test value in each piece and find the sign of f′ there.
Example: f(x) = x³ – 12x. f′(x) = 3x² – 12 = 3(x – 2)(x + 2), zero at –2 and 2.
| Interval | Test value | Sign of f′ | f is |
|---|---|---|---|
| (–∞, –2) | –3 | + | increasing |
| (–2, 2) | 0 | – | decreasing |
| (2, ∞) | 3 | + | increasing |
A function can go up on one stretch and down on the next, as here. For trig functions, use the given interval only, for example (0, 2π).
If f′ is never negative, such as a perfect square 3(x – 1)², f is increasing on the whole line: one point where the slope is 0 does not stop it climbing.
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