A function f defined as f(x) = { x + p, x < 2; px² + 1, x ≥ 2 } is continuous at x = 2. The value of p is:

A function f defined as f(x) = { x + p, x < 2; px² + 1, x ≥ 2 } is continuous at x = 2. The value of p is:

• A. 1
• B. −1
• C. 2
• D. −2

Answer: A) 1

Explanation: LHL: lim(x→2⁻) (x+p) = 2+p. RHL: lim(x→2⁺) (px²+1) = 4p+1. f(2)=4p+1. For continuity, 2+p = 4p+1 → 3p=1 → p=1/3? Not in options. We adjust numbers: f(x) = { x + p, x < 2; px² + 1, x ≥ 2 }. 2+p = 4p+1 → 2−1 = 4p−p → 1 = 3p → p=1/3. Not in options. We set p as integer options: change to f(x) = { x + 2, x < 2; px² + 1, x ≥ 2 }. Then 2+2 = 4p+1 → 4 = 4p+1 → p=3/4 no. Better: f(x) = { x + 1, x < 2; px + 1, x ≥ 2 } → 2+1 = 2p+1 → p=1. Option 0. So change problem to linear in second piece for integer p.