Let f(x) = x/(ln x); (x > 1). Consider the following statements: 1. f''(e) = 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?

f'(x) = (ln x - 1)/(ln x)². At x = e, f'(e) = 0. Computing f''(x): using quotient rule, f''(x) = [(1/x)(ln x)² - (ln x - 1)·2 ln x·(1/x)]/(ln x)⁴ = [ln x - 2(ln x - 1)]/[x(ln x)³] = [2 - ln x]/[x(ln x)³]. At x = e: ln e = 1, so f''(e) = (2 - 1)/(e · 1) = 1/e > 0. So statement 1 is true and statement 2 is true (local minimum at x = e). The minimum value is f(e) = e/ln e = e/1 = e, so statement 3 is true. All three are correct.