Question

Let $f(x)$ be a polynomial of degree $3$ such that $f(3)=21,f^{\prime }(3)=30,f^{\prime \prime }(3)=22$ and $f^{\prime \prime \prime }(3)=6$. Find the value of $f^{\prime }(2)$.

Answer 1

Answer: 11

Let $f(x)=a(x-3{)}^3+b(x-3{)}^2+c(x-3)+d$
$f(3)=21\Rightarrow d=21$
$f^{\prime }(3)=30\Rightarrow c=30$
$f^{\prime \prime }(3)=22\Rightarrow b=11$
$f^{\prime \prime \prime }(3)=6\Rightarrow a=1$
$f(x)=(x-3{)}^3+11(x-3{)}^2+30(x-3)+21$
$\Rightarrow f^{\prime }(x)=3(x-3{)}^2+22(x-3)+30$
$\Rightarrow f^{\prime }(2)=11$