Consider the following statements: 1. f(x) is increasing in the interval (e, \infty) 2. f(x) is decreasing in the interval (1, e) 3. 9\ln 7 \gt 7\ln 9 Which of the statements given above are correct?
Consider the following for the next two (02) items that follow : Let f(x)=\frac{x}{\ln x};(x \gt 1)
- A. 1 and 2 only
- B. 2 and 3 only
- C. 1 and 3 only
- D. 1, 2 and 3 ✓
Correct Answer: D. 1, 2 and 3
Explanation
Find f'(x) = \frac{\ln x \cdot 1 - x \cdot (\frac{1}{x})}{(\ln x)^2} = \frac{\ln x - 1}{(\ln x)^2}. For x \gt e, \ln x \gt 1, so f'(x) \gt 0, meaning f(x) is increasing on (e, \infty) (Statement 1 true). For 1 \lt x \lt e, 0 \lt \ln x \lt 1, so f'(x) \lt 0, meaning f(x) is decreasing on (1, e) (Statement 2 true). Since 9 \gt 7 \gt e, the function is increasing, so f(9) \gt f(7) \implies \frac{9}{\ln 9} \gt \frac{7}{\ln 7} \implies 9\ln 7 \gt 7\ln 9 (Statement 3 true).
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