Question

Let $f\left(x\right)$ is a polynomial of degree $4$, with $f\left(2\right)=-1,f^{\prime }\left(2\right)=0,f^{\prime \prime }\left(2\right)=2,f^{\prime \prime \prime }\left(2\right)=-12,f^{\prime \prime \prime \prime }\left(2\right)=24,$ then the value of $f^{\prime \prime }\left(1\right)$ is -

• A. $24$
• B. $26$
• C. $28$
• D. $30$

Answer 1

The correct option is B $26$

$f\left(x\right)=a(x-2{)}^4+b(x-2{)}^3+c(x-2{)}^2+d(x-2)-1$
$f^{\prime }\left(x\right)=4a(x-2{)}^3+3b(x-2{)}^2+2c(x-2)+d$
$f^{\prime }\left(2\right)=0\Rightarrow d=0$
Similarly $\to f^{\prime \prime }\left(x\right)=12a(x-2{)}^2+6b(x-2)+2c$
$f^{\prime \prime }\left(2\right)=2\Rightarrow 2c=2\Rightarrow c=1$
$f^{\prime \prime \prime }\left(x\right)=24a(x-2)+6b$
$6b=-12\Rightarrow b=-2$
$f^{\prime \prime \prime \prime }\left(x\right)=24a=24\Rightarrow a=1$

Thus, $f^{\prime \prime }\left(x\right)=12(x-2{)}^2-12(x-2)+2$
$f^{\prime \prime }\left(1\right)=26$