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 q ?

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Correct answer: A. -1

For f(x) to be continuous at x=3, we must have _ x 3^- f(x) = _ x 3^+ f(x). 3 - 1 = p(3)^2 + q(3) + 2 2 = 9p + 3q + 2 9p + 3q = 0 3p + q = 0 For f'(x) to be continuous at x=3, the left-hand derivative must equal the right-hand derivative at x=3. f'(x) = 1 for 1 f'(x) = 2px + q for x > 3 Equating the derivatives at x=3: 1 = 2p(3) + q 6p + q = 1 Solving the two equations: 3p + q = 0 6p + q = 1 Subtracting the first equation from the second gives 3p = 1 p = 1 3 . Substituting p into the first equation gives q = -3 ( 1 3 ) = -1. Answer: -1

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