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 ?

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Correct answer: C. 1 3

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 Substituting q = -3p into the above equation: 6p - 3p = 1 3p = 1 p = 1 3 Answer: 1 3

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