The number of intersection points of the graph y = f(x) and the line y = k is the number of real solutions of the equation f(x) = k.
Example ①
Classify the number of real solutions of the equation x³ − 3x = k according to the value of k.
Answer:f(x) = x³−3x. f'(x) = 3x²−3 = 3(x+1)(x−1).
Local maximum f(−1) = 2, local minimum f(1) = −2.
k > 2 or k < −2: 1 solution
k = 2 or k = −2: 2 solutions
−2 < k < 2: 3 solutions
Classify the number of real solutions of the equation x³ − 3x = k according to the value of k.
Answer:f(x) = x³−3x. f'(x) = 3x²−3 = 3(x+1)(x−1).
Local maximum f(−1) = 2, local minimum f(1) = −2.
k > 2 or k < −2: 1 solution
k = 2 or k = −2: 2 solutions
−2 < k < 2: 3 solutions
Example ②
For a > 0, find the extreme values of f(x) = x³ − 3a²x + 2.
Answer:f'(x) = 3x²−3a² = 3(x+a)(x−a).
Local maximum f(−a) = 2a³+2, local minimum f(a) = −2a³+2
For a > 0, find the extreme values of f(x) = x³ − 3a²x + 2.
Answer:f'(x) = 3x²−3a² = 3(x+a)(x−a).
Local maximum f(−a) = 2a³+2, local minimum f(a) = −2a³+2
Key Points
- f(x) = k ⟺ the number of intersection points of y = f(x) and y = k
- The number of solutions is determined by the positions of the local maximum, local minimum, and k for a cubic function
- There are 3 solutions when the local maximum value > k > the local minimum value
Practice Problems
- Find the range of a for which x³ − 3x − a = 0 has three real solutions.
- Find the extreme values of f(x) = x³ + 3x² − 1, and find the number of real solutions of x³ + 3x² − 1 = 0.
- Find the condition on a for f(x) = x³ − ax to have an extreme value for x > 0.
Answers and Explanations
- Answer:−2 < a < 2
Explanation: Compare with the local maximum 2 and local minimum −2 of f(x)=x³−3x. f(x)=a has 3 solutions ↔ −2 < a < 2. - Answer:Local maximum f(−2)=3, local minimum f(0)=−1. Intersections of the graph with y=0: since −1<0<3, there are 3.
Explanation: f'=3x²+6x=3x(x+2). There is a local maximum at x=−2 and a local minimum at x=0. - Answer:a > 0
Explanation: f'=3x²−a. An extreme value for x>0 ↔ f'=0 has a positive solution ↔ a>0.