Question

Let $f : R \to R$ be a function such that $f(x) = x^3 + x^2 f'(1) + xf''(2)+f'''(3), x \in R$. Then $f(2)$ equal :

• A. 8
• B. -2
• C. -4
• D. 30

Answer 1

Answer: (B)

$f(x) = x^3 + x^2 f'(1) + xf''(2) + f'''(3)$
$\Rightarrow \; ƒ'(x) = 3x2 + 2xƒ'(1) + ƒ''(x) $ .....(1)
$\Rightarrow \; ƒ''(x) = 6x + 2ƒ'(1)$.....(2)
$\Rightarrow \; ƒ'''(x) = 6$ .....(3)
put x = 1 in equation (1) :
$ƒ'(1) = 3 + 2ƒ'(1) + ƒ''(2)$ .....(4)
put x = 2 in equation (2) :
$ƒ''(2) = 12 + 2ƒ'(1)$ .....(5)
from equation (4) & (5) :
$-3 - ƒ'(1) = 12 + 2ƒ'(1)$
$\Rightarrow \; 3ƒ'(1) = -15$
$\Rightarrow \; ƒ'(1) = -5 Þ ƒ''(2) = 2$ ....(2)
put $x = 3$ in equation $(3)$ :
$ƒ'''(3) = 6$
$\therefore \; ƒ(x) = x^3 - 5x^2 + 2x + 6$
$ƒ(2) = 8 - 20 + 4 + 6 = -2$