Consider the following statements: 1. f^{\prime\prime}(e)=\frac{1}{e} 2. f(x) attains local minimum value at x=e 3. A local minimum value of f(x) is e 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
From f'(x) = \frac{\ln x - 1}{(\ln x)^2}, set f'(x) = 0 to get \ln x = 1 \implies x = e. Taking the second derivative, f''(x) = \frac{(\ln x)^2(1/x) - (\ln x - 1)2(\ln x)(1/x)}{(\ln x)^4}. At x=e, f''(e) = \frac{(1)(1/e) - 0}{1} = \frac{1}{e}. Since f''(e) \gt 0, x=e is a point of local minimum. The minimum value is f(e) = \frac{e}{\ln e} = e. Thus, all three statements are true.
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