Rising and falling
A function is:
- increasing on an interval if f′(x) > 0 there (the graph rises left to right),
- decreasing if f′(x) < 0 (the graph falls).
Step 1: Find critical numbers
A critical number is where f′(x) = 0 or f′(x) is undefined (but f(x) is defined).
Example: f(x) = x³ − 6x² + 9x + 1
- f′(x) = 3x² − 12x + 9 = 3(x − 1)(x − 3)
- Critical numbers: x = 1 and x = 3
Step 2: Make a sign chart
Split the number line at the critical numbers into intervals. Test a value in each.
| Interval | Test x | f′(x) = 3(x − 1)(x − 3) | f |
|---|---|---|---|
| x < 1 | 0 | 3(−1)(−3) = 9 > 0 | increasing |
| 1 < x < 3 | 2 | 3(1)(−1) = −3 < 0 | decreasing |
| x > 3 | 4 | 3(3)(1) = 9 > 0 | increasing |
This is a sign chart.
Step 3: State the intervals
f is increasing on (−∞, 1) and (3, ∞); decreasing on (1, 3).
The first derivative test
At a critical number:
| Sign change of f′ | Result |
|---|---|
| + to − | local maximum |
| − to + | local minimum |
| no change | neither |
From the example:
- x = 1: + to − → local max f(1) = 5
- x = 3: − to + → local min f(3) = 1
Example with no change
f(x) = x³: f′(x) = 3x² ≥ 0. Critical number x = 0, but f′ doesn’t change sign → no max or min (just a flat spot).
Example with an undefined derivative
f(x) = x^(2/3): f′(x) = (2/3)x^(−1/3), undefined at 0. Sign: − for x < 0, + for x > 0 → local min at x = 0 (a cusp).
In context
Example: Revenue R(x) = −2x² + 80x. R′(x) = −4x + 80 = 0 → x = 20.
R′ > 0 for x < 20 (revenue rising), R′ < 0 for x > 20 (falling). Revenue peaks at x = 20.
Worked examples
Example 1: f(x) = x² − 6x. Where increasing? f′ = 2x − 6 > 0 → x > 3.
Example 2: f(x) = −x³ + 3x. Critical numbers? −3x² + 3 = 0 → x = ±1.
Example 3: For example 2, classify x = 1. f′(0) = 3 > 0, f′(2) = −9 < 0 → local max.
Common mistakes
- Testing at the critical numbers instead of between them.
- Forgetting points where f′ is undefined.
- Calling every critical number a max or min.
Try it yourself
- f(x) = x² + 4x. Intervals of increase and decrease?
- f(x) = 2x³ − 6x. Critical numbers?
- Classify them using the first derivative test.
- f(x) = x⁴. Is x = 0 a max, min or neither?
Answers: 1) decreasing for x < −2, increasing for x > −2 2) ±1 3) local max at x = −1, local min at x = 1 4) local (and absolute) minimum
Quick recap
- f′ > 0: increasing; f′ < 0: decreasing.
- Find critical numbers, make a sign chart across the intervals.
- The first derivative test classifies local maximums and minimums.