Question

If $f\left(x\right)+2f(1-x)=x^2+2$; $\forall x\in R$, then $f\left(x\right)$ is given by

• A. $\frac{(x-1{)}^2}{3}$
• B. $-\frac{(x-2{)}^2}{3}$
• C. $x^2-1$
• D. $x^2-2$

Answer 1

Answer: (B)

$-\frac{(x-2{)}^2}{3}$

$\Rightarrow$$f\left(x\right)+2f(1-x)=x^2+2\cdots \cdots \left(i\right)$

Put $x=1-x$

$\Rightarrow$$f(1-x)+2f\left(x\right)=(1-x{)}^2+2$

Multiply by $2$

$\Rightarrow$$2f(1-x)+4f\left(x\right)=2(1-x{)}^2+4\cdots \cdots (ii)$

Subtract $\left(ii\right)$ from $\left(i\right)$

$-3f\left(x\right)=x^2-2-2(1-x{)}^2$

$-3f\left(x\right)=x^2-2-2x^2+4x$

$-3f\left(x\right)=-(x^2-4x+4)$

$f\left(x\right)=-\frac{(x-2{)}^2}{3}$