Question

If $f\left(x\right)+2f(1-x)=6x\forall x\in R,$ then $\frac{f(-x)-f\left(x\right)}{2}$ equals

• A. $x$
• B. $3x$
• C. $6x$
• D. $12x$

Answer 1

The correct option is C $6x$

$f\left(x\right)+2f(1-x)=6x\dots \left(1\right)$
Replacing $x$ by $1-x,$ we get
$f(1-x)+2f\left(x\right)=6(1-x)$
$\Rightarrow 2f(1-x)+4f\left(x\right)=12(1-x)\dots \left(2\right)$

From $\left(2\right)-\left(1\right),$
$3f\left(x\right)=12-18x$
$\Rightarrow f\left(x\right)=4-6x$
$\therefore f(-x)-f\left(x\right)=12x$
$\therefore \frac{f(-x)-f\left(x\right)}{2}=6x$