Single correct option
f(x) is a quadratic polynomial such that f(x + 1) + f(x − 1) = 2x² − 4x + 6 for all real x. What is f(3)?
- A. 3
- B. 5
- C. 7
- D. 9
Show answer and solution
Answer: (B) 5
Let f(x) = ax² + bx + c.
f(x + 1) + f(x − 1) = a[(x + 1)² + (x − 1)²] + b[2x] + 2c = 2ax² + 2bx + 2a + 2c.
Compare with 2x² − 4x + 6: 2a = 2 ⇒ a = 1; 2b = −4 ⇒ b = −2; 2a + 2c = 6 ⇒ c = 2.
f(x) = x² − 2x + 2, so f(3) = 9 − 6 + 2 = 5.
f(x + 1) + f(x − 1) = a[(x + 1)² + (x − 1)²] + b[2x] + 2c = 2ax² + 2bx + 2a + 2c.
Compare with 2x² − 4x + 6: 2a = 2 ⇒ a = 1; 2b = −4 ⇒ b = −2; 2a + 2c = 6 ⇒ c = 2.
f(x) = x² − 2x + 2, so f(3) = 9 − 6 + 2 = 5.
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