Question

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

Answer 1

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