Consider the following statements: Statement-I: The function f(x) = x^3 + 128 x has a minimum 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?

Step-by-step solution

Given f(x) = x^3 + 128 x = x^2 + 128 x Differentiating with respect to x , we get: f'(x) = 2x - 128 x^2 = 2(x^3 - 64) x^2 For critical points, f'(x) = 0 x^3 - 64 = 0 x = 4 At x = 4 , f(4) = 4^3 + 128 4 = 192 4 = 48 Now, checking the sign of f'(x) around x = 4 : For x For x > 4 , x^3 > 64 f'(x) > 0 (positive) Since f'(x) changes sign from negative to positive as x increases through 4 , x = 4 is a point of local minimum. The minimum value is 48 . Thus, Statement-I is correct. Statement-II claims that f'(x) changes si