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 :

Step-by-step solution

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