Question

If $f(x)=3x^{2}+2xf^{\prime}(1)+f^{\prime\prime}(2)$, then $f(x)=..........$

• A. $3x^{2}+8x+4$
• B. $3x^{2}+12x+12$
• C. $3x^{2}-12x+6$
• D. $3x^{2}-18x+5$

Hint:

Constants like $f'(a)$ in an equation can be treated as unknowns and solved by differentiating the original equation.

Answer 1

The Correct Option is C

Step 1: Understanding the Question:
The coefficients of $f(x)$ are its own derivatives evaluated at specific points. We solve for these constant values.

Step 2: Key Formula or Approach:
Let $f'(1) = A$ and $f''(2) = B$. Then $f(x) = 3x^2 + 2Ax + B$.
Differentiate to relate $A$ and $B$.

Step 3: Detailed Explanation:
$f(x) = 3x^2 + 2Ax + B$
$f'(x) = 6x + 2A$
$f''(x) = 6$.
Now evaluate at the specific points:
1. $f'(1) = A \implies 6(1) + 2A = A \implies A = -6$.
2. $f''(2) = B \implies 6 = B \implies B = 6$.
Substitute $A$ and $B$ back into the original equation:
$f(x) = 3x^2 + 2(-6)x + 6 = 3x^2 - 12x + 6$.

Step 4: Final Answer:
The function is $f(x) = 3x^2 - 12x + 6$.