Question

Let $f\left(x\right)+2f(1-x)=x^2+2\forall x\in R$ then the interval in which $f\left(x\right)$ increases is

• A. $(-\infty ,-2)$
• B. $(-\infty ,\infty )$
• C. $(2,\infty )$
• D. None of these

Answer 1

Answer: (C)

$(2,\infty )$

$f\left(x\right)+2f(1-x)=x^2+\mathrm{2...}\left(i\right)$
$\Rightarrow f(1-x)+2f\left(x\right)={(1-x)}^2+\mathrm{2...}\left(ii\right)$
replacing $x\to 1-x$
multiplying (ii) by $2$ then subtracting (1) we get
$3f\left(x\right)=x^2-4x+4=(x-2)$
$\Rightarrow f\left(x\right)=\frac{{(x-2)}^2}{3}$
differentiating w.r.to $x$ both sides we get
$f^{\prime }\left(x\right)=\frac{2(x-2)}{3}>0\forall (2,\infty )$