Let f(x) = ax(x-1) for x 3. Given that f(x) is continuous for all x but not differentiable at x = 1. Further f'(x) is continuous at x = 3. What is the value of q?

Question

Let f(x) = ax(x-1) for x < 1; x - 1 for 1 ≤ x ≤ 3; px² + qx + 2 for x > 3. Given that f(x) is continuous for all x but not differentiable at x = 1. Further f'(x) is continuous at x = 3. What is the value of q?

Accepted Answer

-1. From the conditions: continuity at x = 3 gives 9p + 3q + 2 = 2, so 9p + 3q = 0, hence q = -3p. Differentiability at x = 3 gives 6p + q = 1. Substituting q = -3p: 6p - 3p = 1, so p = 1/3, and q = -3(1/3) = -1.

Let f(x) = ax(x-1) for x < 1; x - 1 for 1 ≤ x ≤ 3; px² + qx + 2 for x > 3. Given that f(x) is continuous for all x but not differentiable at x = 1. Further f'(x) is continuous at x = 3. What is the value of q?

Explanation

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