By examining the sign of f'(x), we can determine whether a function is increasing or decreasing, its extreme values, and the shape of its graph.
Table of Increase and Decrease
- Intervals where f'(x) > 0: f(x) is increasing
- Intervals where f'(x) < 0: f(x) is decreasing
- Points where f'(x) = 0: candidates for extreme values
If f'(a) = 0 and the signs before and after are + → −, there is a local maximum; if they are − → +, there is a local minimum.
Example ①
Create a table of increase and decrease for f(x) = x³ − 3x, and find its extreme values.
Answer:f'(x) = 3x² − 3 = 3(x+1)(x−1).
f'(x) = 0 → x = ±1
Table of increase and decrease:
x: … −1 … 1 …
f': … + 0 − 0 +
f: ↗ local maximum 2 ↘ local minimum −2 ↗
Local maximum f(−1) = 2, local minimum f(1) = −2
Create a table of increase and decrease for f(x) = x³ − 3x, and find its extreme values.
Answer:f'(x) = 3x² − 3 = 3(x+1)(x−1).
f'(x) = 0 → x = ±1
Table of increase and decrease:
x: … −1 … 1 …
f': … + 0 − 0 +
f: ↗ local maximum 2 ↘ local minimum −2 ↗
Local maximum f(−1) = 2, local minimum f(1) = −2
Key Points
- Factor f'(x) and examine its sign
- A table of increase and decrease is essential (show the sign changes clearly)
- Even if f'(a) = 0, there may be no extreme value (for example, x=0 for f'(x) = x³)
Practice Problems
- Create a table of increase and decrease for f(x) = x³ − 6x² + 9x, and find its extreme values.
- Find the local maximum and local minimum values of f(x) = −x³ + 3x².
- Find the extreme values of f(x) = x⁴ − 2x².
Answers and Explanations
- Answer:f'(x)=3x²−12x+9=3(x−1)(x−3). Local maximum f(1)=4, local minimum f(3)=0
Explanation: +→− at x=1 (local maximum), −→+ at x=3 (local minimum). - Answer:Local maximum f(2)=4, local minimum f(0)=0
Explanation: f'=−3x²+6x=−3x(x−2). At x=0, −→+ (local minimum); at x=2, +→− (local maximum). - Answer:Local minima f(±1)=−1 at x=±1, local maximum f(0)=0 at x=0
Explanation: f'=4x³−4x=4x(x²−1)=4x(x+1)(x−1). Check the sign changes.