Question

If \[ f(x) = \begin{cases} 6\beta - 3x, & \text{if } -4 \leq x<-2,
4x + 1, & \text{if } -2 \leq x \leq 2, \end{cases} \] is continuous on \( [-4, 2] \), then \( \alpha + \beta = \)

• A. \( -\frac{7}{6} \)
• B. \( \frac{4}{7} \)
• C. \( -\frac{4}{7} \)
• D. \( \frac{7}{6} \)

Hint:

To ensure continuity of a piecewise function, set the left-hand and right-hand limits equal to each other at the boundary points and solve for the unknowns.

Answer 1

The Correct Option is A

Step 1: Apply continuity condition.
For the function to be continuous on \( [-4, 2] \), the left-hand limit and right-hand limit at \( x = -2 \) must be equal. Therefore, we need to equate the two expressions for \( f(x) \) at \( x = -2 \): \[ 6\beta - 3(-2) = 4(-2) + 1. \]
Step 2: Solve for \( \alpha + \beta \).
Simplifying both sides, we get: \[ 6\beta + 6 = -8 + 1. \] Solving for \( \beta \), we get \( \beta = -\frac{7}{6} \).
Step 3: Conclusion.
Thus, \( \alpha + \beta = -\frac{7}{6} \), making option (A) the correct answer.