The sign of f'(x) tells us how f(x) increases or decreases.
- On intervals where f'(x) > 0, f(x) is increasing
- On intervals where f'(x) < 0, f(x) is decreasing
Example ① Investigate the increase and decrease of f(x) = x³ − 3x.
Solution: f'(x) = 3x²−3 = 3(x−1)(x+1).
Variation table: f'>0 (increasing) when x<−1; f'<0 (decreasing) when −10 (increasing) when x>1.
Therefore, there is a local maximum value of 2 at x=−1 and a local minimum value of −2 at x=1.
Solution: f'(x) = 3x²−3 = 3(x−1)(x+1).
Variation table: f'>0 (increasing) when x<−1; f'<0 (decreasing) when −1
Therefore, there is a local maximum value of 2 at x=−1 and a local minimum value of −2 at x=1.
Key Point
- Solve f'(x) = 0 → candidate points
- Check sign changes in the variation table → determine local maxima and minima
Practice Problem
- Find the local extrema of f(x) = x³ − 6x² + 9x.
Answer and Explanation
- Answer:Local maximum value 4 at x=1; local minimum value 0 at x=3
f'=3x²−12x+9=3(x−1)(x−3).