Let $\mathrm{f}: \mathrm{R} \rightarrow \mathrm{R}$ 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 :
Correct Option: , 2
$f(x)=x^{3}+x^{2} f^{\prime}(1)+x f^{\prime \prime}(2)+f^{\prime \prime \prime}(3)$
$\Rightarrow f^{\prime}(x)=3 x^{2}+2 x f^{\prime}(1)+f^{\prime \prime}(x)$ $\ldots \ldots(1)$
$\Rightarrow f^{\prime \prime}(x)=6 x+2 f^{\prime}(1)$ $\ldots \ldots(2)$
$\Rightarrow f^{\prime \prime \prime}(x)=6$ $\ldots . .(3)$
put $x=1$ in equation (1):
$f^{\prime}(1)=3+2 f^{\prime}(1)+f^{\prime \prime}(2)$ $\ldots . .(4)$
put $x=2$ in equation (2):
$f^{\prime \prime}(2)=12+2 f^{\prime}(1)$ $\ldots . .(5)$
from equation (4) \& (5) :
$-3-f^{\prime}(1)=12+2 f^{\prime}(1)$
$\Rightarrow 3 f^{\prime}(1)=-15$
$\Rightarrow f^{\prime}(1)=-5 \Rightarrow f^{\prime \prime}(2)=2 \quad \ldots .(2)$
put $x=3$ in equation (3):
$f^{\prime \prime \prime}(3)=6$
$\therefore f(x)=x^{3}-5 x^{2}+2 x+6$
$f(2)=8-20+4+6=-2$