The second derivative f''(x) shows the graph’s concavity.
- f''(x) > 0: concave upward (∪)
- f''(x) < 0: concave downward (∩)
- A point where f''(x) = 0 and the sign changes is called an inflection point
Example ① Find the concavity and inflection point of f(x) = x³ − 3x².
Solution: f'=3x²−6x, f''=6x−6. f''=0 ⇔ x=1.
For x<1: f''<0 (concave downward); for x>1: f''>0 (concave upward).
Inflection point: (1, −2)
Solution: f'=3x²−6x, f''=6x−6. f''=0 ⇔ x=1.
For x<1: f''<0 (concave downward); for x>1: f''>0 (concave upward).
Inflection point: (1, −2)
Key Point
- The sign of f''(x) → concavity
- At an inflection point, the direction of the graph’s curvature reverses
Practice Problem
- Find the inflection points of f(x) = x⁴ − 6x².
Answer and Explanation
- Answer:(1, −5) and (−1, −5)
f''=12x²−12=12(x−1)(x+1).