Let f(x)=x^3+x^2f^'(1)+2xf^''(2)+f^'''(3), x R.

Question

Let $f(x)={x}^{3}+{x}^{2}{f}^{'}(1)+2x{f}^{''}(2)+{f}^{'''}(3)$, $x\in R$. Then the value of ${f}^{'}(5)$ is :

Accepted Answer

$\begin{gathered} {f}^{'}(1)=a \ {f}^{'}(2)=b \ {f}^{'}(3)=c \ f(x)={x}^{3}+a{x}^{2}+bx+c \ {f}^{'}(x)=3{x}^{2}+2ax+b \ {f}^{'}(x)=a=3+2a+b\Rightarrow a+b=3\ldots ..(1) \ {f}^{''}(x)=6x+2a \ {f}^{''}(2)=12+2a=\frac{b}{2}\Rightarrow 4a-b=-24 \ {f}^{''}(x)=6 \ {f}^{''}(3)=c=6 \ \Rightarrow a=\frac{-27}{5},b=\frac{12}{5} \ {f}^{'}(5)=75+10a+b \ =75-54+\frac{12}{5}=21+\frac{12}{5}=\frac{117}{5} \end{gathered}$

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