Question

If $f\left(x\right)+2f\left(1-x\right)=6x\left(\forall x\in R\right)$ , then the value of $\frac{3}{4}\left(\frac{f\left(8\right)}{f^{{}^{\prime }}\left(1\right)}\right)$ is equal to

Answer 1

Answer: 5.5

In given equation substituting $x$ by $1-x$ , we get,
$f(1-x)+2f(1-(1-x))=6(1-x)$
From these two equations, we get,
$f\left(x\right)+2\left\{6-6x-2f\left(x\right)\right\}=6x$
$-3f\left(x\right)+12-12x=6x$
$3f\left(x\right)=12-18x\Rightarrow f\left(x\right)=4-6x$
$f\left(8\right)=-44,f^{{}^{\prime }}\left(1\right)=-6$
$\frac{f\left(8\right)}{f^{{}^{\prime }}\left(1\right)}=\frac{-44}{-6}=\frac{22}{3}$