Passage: Let f(x)= cases ax(x-1), & x 3 cases Given that f(x) is continuous for all x but not differentiable at x=1 . Further f'(x) is continuous at x=3 . Question: What is the value of p ?

Step-by-step solution

For f(x) to be continuous at x=3 , the left-hand limit and right-hand limit at x=3 must be equal. _ x 3^- f(x) = 3 - 1 = 2 _ x 3^+ f(x) = p(3)^2 + q(3) + 2 = 9p + 3q + 2 Equating them, we get: 9p + 3q + 2 = 2 9p + 3q = 0 q = -3p The derivative of the function for x (1, 3) is f'(x) = 1 , and for x > 3 is f'(x) = 2px + q . Since f'(x) is continuous at x=3 , the left-hand derivative and right-hand derivative at x=3 must be equal. _ x 3^- f'(x) = 1 _ x 3^+ f'(x) = 2p(3) + q = 6p + q Equating them, we get: 6p + q = 1 Su