Problems involving extreme values of functions containing parameters, proofs of inequalities, and so on.
Example ① The range of a for which f(x) = x³ − 3ax + 1 has extreme values.
Solution:f'(x) = 3x²−3a = 3(x²−a)。
Having extreme values ⇔ f'(x)=0 has two distinct solutions ⇔ a > 0
Solution:f'(x) = 3x²−3a = 3(x²−a)。
Having extreme values ⇔ f'(x)=0 has two distinct solutions ⇔ a > 0
Example ② Prove that log x ≦ x − 1 for x > 0.
Solution:Let f(x) = x − 1 − log x. f'(x) = 1 − 1/x.
The minimum value is 0 at x = 1. Therefore, f(x) ≧ 0, that is, log x ≦ x − 1。
Solution:Let f(x) = x − 1 − log x. f'(x) = 1 − 1/x.
The minimum value is 0 at x = 1. Therefore, f(x) ≧ 0, that is, log x ≦ x − 1。
Key Point
- To prove the inequality f(x) ≧ 0: regard the left side as a function and show that its minimum value is at least 0
Practice Problems
- Prove that e^x > 1 + x for x > 0。
Solutions and Explanations
- Solution:f(x)=e^x−1−x、f'=e^x−1>0（x>0）、and f(0)=0, so f(x)>0。