For the following three (03) items: Consider the function f(x) = x|x| . Consider the following statements: I. The function is increasing in the interval (- , ) . II. The function is differentiable at x = 0 . Which of the statements given above is/are correct?

Step-by-step solution

The given function is f(x) = x|x| . This can be redefined as: f(x) = x^2 for x 0 f(x) = -x^2 for x Differentiating with respect to x , we get: f'(x) = 2x for x > 0 f'(x) = -2x for x At x = 0 , the left-hand derivative is _ h 0^- -h^2 - 0 h = 0 and the right-hand derivative is _ h 0^+ h^2 - 0 h = 0 . Since the left-hand derivative equals the right-hand derivative, f(x) is differentiable at x = 0 . Thus, statement II is correct. Also, f'(x) = 2|x| 0 for all x (- , ) . Since f'(x) > 0 for all x 0 and f(x) is continuou