Question

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

• A. 8
• B. -2
• C. -4
• D. 30

Answer 1

Answer: (B)

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