Question

If \( f(x) = x^2 + 2x f'(1) + f''(2) \) for all \( x \), then \( f(0) \) is equal to:

• A. \( 4 \)
• B. \( 3 \)
• C. \( 2 \)
• D. \( 1 \)
• E. \( 0 \)

Hint:

For given functional equations, differentiate step-by-step to find required derivatives.

Answer 1

The Correct Option is C

Step 1: Find \( f'(x) \) Given: \[ f(x) = x^2 + 2x f'(1) + f''(2). \] Differentiate both sides: \[ f'(x) = 2x + 2f'(1). \] Substituting \( x = 1 \): \[ f'(1) = 2(1) + 2f'(1). \] \[ f'(1) - 2f'(1) = 2. \] \[ - f'(1) = 2 \Rightarrow f'(1) = -2. \] Step 2: Find \( f''(x) \) Differentiating again: \[ f''(x) = 2. \] So: \[ f''(2) = 2. \] Step 3: Compute \( f(0) \) \[ f(0) = 0^2 + 2(0)(-2) + 2 = 2. \] Thus, the correct answer is (C) 2.