Question:
If \( f(x) = x^2 + 2x f'(1) + f''(2) \) for all \( x \), then \( f(0) \) is equal to:
If \( f(x) = x^2 + 2x f'(1) + f''(2) \) for all \( x \), then \( f(0) \) is equal to:
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For given functional equations, differentiate step-by-step to find required derivatives.
Updated On: Apr 7, 2026
- \( 4 \)
- \( 3 \)
- \( 2 \)
- \( 1 \)
- \( 0 \)
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The Correct Option is C
Solution and Explanation
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.
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