Question

The value of constant \( c \) that makes the function \( f \) defined by \(f(x) = \ x^2 - c^2\), \(\&\ \text{if } x<4\)\(cx + 20, \&\ \text{if } x \geq 4\)   continuous for all real numbers is:

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

Hint:

To ensure continuity of piecewise functions, equate the left-hand limit and right-hand limit at the boundary point(s).

Answer 1

The Correct Option is A

Step 1: Continuity condition at \( x = 4 \). For \( f(x) \) to be continuous at \( x = 4 \), the left-hand limit (\( LHL \)) must equal the right-hand limit (\( RHL \)) and the value of \( f(4) \). \[ LHL = \lim_{x \to 4^-} (x^2 - c^2) = 4^2 - c^2 = 16 - c^2 \] \[ RHL = \lim_{x \to 4^+} (cx + 20) = 4c + 20 \] Equating \( LHL \) and \( RHL \): \[ 16 - c^2 = 4c + 20 \] Step 2: Solving the equation. Rearranging: \[ c^2 + 4c + 4 = 0 \quad \Rightarrow \quad (c + 2)^2 = 0 \quad \Rightarrow \quad c = -2 \] Conclusion: Thus, the required value of \( c \) is \( -2 \), which corresponds to option \( \mathbf{(A)} \).