What is the value of p ?

For the next two (02) items that follow : Let \( f(x) = \begin{cases} ax(x-1), & x < 1 \\ x-1, & 1 \le x \le 3 \\ px^2 + qx + 2, & x > 3 \end{cases} \) Given that \( f(x) \) is continuous for all x but not differentiable at x = 1. Further \( f'(x) \) is continuous at x = 3.

  1. A. -1
  2. B. -\frac{1}{3}
  3. C. \frac{1}{3} ✓
  4. D. 1

Correct Answer: C. \frac{1}{3}

Explanation

Since \( f(x) \) is continuous at x = 3, equating limits: \( 3 - 1 = p(3)^2 + q(3) + 2 \implies 9p + 3q = 0 \implies 3p + q = 0 \). Since \( f'(x) \) is continuous at x = 3, equating derivatives: \( 1 = 2p(3) + q \implies 6p + q = 1 \). Subtracting equations yields \( 3p = 1 \implies p = 1/3 \).

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