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?
D. Neither I nor II
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) = _ x 1^- d dx (x^3) = _ x 1^- 3x^2 = 3(1)^2 = 3 Right Hand Derivative (RHD) = _ x 1^+ d dx (x^2) = _ x 1^+ 2x = 2(1) = 2 Since LHD RHD, f(x) is not differentiable at x = 1. Statement II is incorrect. Both statements are incorrect. Answer: Neither I nor II
Related: Mathematics — Continuity and Differentiability · All PYQ Banks