Passage: Let f(x)= (x^2) and g(x)=x|x| for |x| Question: If q(x)=f g(x) , then which of the following statements is/are correct ? I. q(x) is continuous at x=0 . II. q(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 composite function q(x) is given by: q(x) = f(g(x)) = f(x|x|) = ((x|x|)^2) Since (x|x|)^2 = x^2|x|^2 = x^4 , we have: q(x) = (x^4) Checking continuity at x=0 : _ x 0 q(x) = _ x 0 (x^4) = 0 Also, q(0) = (0) = 0 . Since _ x 0 q(x) = q(0) , q(x) is continuous at x=0 . Statement I is correct. Checking differentiability at x=0 : q'(0) = _ x 0 q(x) - q(0) x - 0 = _ x 0 (x^4) x q'(0) = _ x 0 ( (x^4) x^4 x^3 ) = 1 0 = 0 Since the limit exists and is finite, q(x) is differentiable at