Passage: Let f(x)= (x^2) and g(x)=x|x| for |x| Question: If p(x)=f(x)g(x), then which of the following statements is/are correct ? I. p(x) is continuous at x=0. II. p(x) is differentiable at x=0. Select the answer using the code given below :
C. Both I and II
Given f(x) = (x^2) and g(x) = x|x|. The function p(x) is given by p(x) = f(x)g(x) = x|x| (x^2). We can redefine p(x) as: p(x) = x^2 (x^2) for x 0 p(x) = -x^2 (x^2) for x Checking continuity at x = 0: _ x 0^+ p(x) = _ x 0^+ x^2 (x^2) = 0 _ x 0^- p(x) = _ x 0^- -x^2 (x^2) = 0 p(0) = 0 Since _ x 0^+ p(x) = _ x 0^- p(x) = p(0), p(x) is continuous at x = 0. Statement I is correct. Checking differentiability at x = 0: Right Hand Derivative (RHD) at x = 0: _ h 0^+ p(h) - p(0) h = _ h 0^+ h^2 (h^2) - 0 h = _ h 0^+ h (h^2) = 0 Left Hand Derivative (LHD) at x = 0: _ h 0^- p(h) - p(0) h = _ h 0^- -h^2 (h^2) - 0 h = _ h 0^- -h (h^2) = 0 Since RHD = LHD = 0, p(x) is differentiable at x = 0. Statement II is correct. Both statements I and II are correct. Answer: Both I and II
Related: Mathematics — Continuity and Differentiability · All PYQ Banks