Question

If
\[ f(x) = \begin{cases} 1 + x & \text{if } x < 0, \\ (1 - x)(px + q) & \text{if } x \geq 0, \end{cases} \] satisfies the assumptions of Rolle’s theorem in the interval \( [-1, 1] \), then the ordered pair \( (p, q) \) is

• A. \( (1, 1) \)
• B. \( (2, 1) \)
• C. \( (1, 0) \)
• D. \( (0, 1) \)

Hint:

For Rolle's Theorem, ensure the function is continuous and differentiable on the interval and check the boundary conditions.

Answer 1

The Correct Option is C

Step 1: Applying Rolle’s Theorem. 
Rolle’s Theorem requires that \( f(-1) = f(1) \). For \( f(x) \) to satisfy this condition, we need to find \( p \) and \( q \) such that the function is continuous and differentiable at \( x = 0 \). 

Step 2: Solving for \( p \) and \( q \). 
By substituting \( x = 0 \) into both cases, we can solve for \( p \) and \( q \). The condition \( f(-1) = f(1) \) leads to the solution \( (p, q) = (1, 0) \). 

Step 3: Conclusion. 
Thus, the correct answer is (C).