Question

If $f\left(x\right)=(x-2{)}^2(x+1{)}^3(x-3{)}^4,$ then the value of $f^{\prime \prime \prime }\left(3\right)+f^{\prime \prime }(-1)+f^{\prime }\left(2\right)$ is

• A. $0$
• B. $2$
• C. $-4$
• D. Does not exist

Answer 1

The correct option is A $0$

$(x-2)$ is a factor of $f\left(x\right)$ repeated $2$ times
$\Rightarrow f^{\prime }\left(2\right)=0$

$(x+1)$ is a factor of $f\left(x\right)$ repeated $3$ times
$\Rightarrow f^{\prime \prime }(-1)=0$

$(x-3)$ is a factor of $f\left(x\right)$ repeated $4$ times
$\Rightarrow f^{\prime \prime \prime }\left(3\right)=0$
$\therefore f^{\prime \prime \prime }\left(3\right)+f^{\prime \prime }(-1)+f^{\prime }\left(2\right)=0$