Question

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

Answer 1

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