Consider the following statements: Statement-I: The function f(x)=\frac{x^{3}+128}{x} has a <strong>MINIMUM</strong> value 48 at x = 4. Statement-II: As x increases through 4, f'(x) changes sign from positive to negative. Which one of the following is correct in respect of the above statements?
- A. Both Statement-I and Statement-II are correct and Statement-II explains Statement-I
- B. Both Statement-I and Statement-II are correct but Statement-II does not explain Statement-I
- C. Statement-I is correct but Statement-II is not correct ✓
- D. Statement-I is not correct but Statement-II is correct
Correct Answer: C. Statement-I is correct but Statement-II is not correct
Explanation
We can rewrite f(x) = x^2 + \frac{128}{x}. Then f'(x) = 2x - \frac{128}{x^2}. Setting f'(x) = 0 gives 2x^3 = 128 \implies x = 4. Evaluating f(4) = 16 + 32 = 48. For x \lt 4, f'(x) \lt 0 and for x \gt 4, f'(x) \gt 0. Thus, the sign of f'(x) changes from negative to positive, confirming a local minimum. Statement-I is correct. Statement-II is incorrect because it states the sign changes from positive to negative.
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