Consider the following for the two (02) items that follow: Let f(x) = cases x^3, & x^2 Consider the following statements: I. The function is continuous at x = -1 . II. The function is differentiable at x = 1 . Which of the statements given above is/are correct?

Step-by-step solution

The given function can be rewritten as: f(x) = cases x^2, & x -1 x^3, & -1 Checking continuity at x = -1 : Left Hand Limit (LHL) = _ x -1^- x^2 = (-1)^2 = 1 Right Hand Limit (RHL) = _ x -1^+ x^3 = (-1)^3 = -1 Since LHL RHL, f(x) is not continuous at x = -1 . Statement I is incorrect. Checking differentiability at x = 1 : First, check continuity at x = 1 : LHL = _ x 1^- x^3 = 1 RHL = _ x 1^+ x^2 = 1 f(1) = 1^2 = 1 The function is continuous at x = 1 . Now, check the derivatives at x = 1 : Left Hand Derivative (LHD)