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: Concept
Let $f'(1) = A$ and $f''(2) = B$. Then $f(x) = 3x^2 + 2Ax + B$.

Step 2: Analysis
$f'(x) = 6x + 2A \implies f'(1) = 6 + 2A$.
So $A = 6 + 2A \implies A = -6$.
$f''(x) = 6 \implies f''(2) = 6$. So $B = 6$.

Step 3: Calculation
Substitute $A = -6$ and $B = 6$ into $f(x)$:
$f(x) = 3x^2 + 2(-6)x + 6 = 3x^2 - 12x + 6$.

Step 4: Conclusion
Hence, the function is $3x^2 - 12x + 6$. Final Answer: (C)