Question

Range of the function $f\left(x\right)=x^2+\frac{1}{x^2+1}$, is

• A. $[1,\infty )$
• B. $[2,\infty )$
• C. $[\frac{3}{2},\infty )$
• D. $(-\infty ,\infty )$

Answer 1

The correct option is A $[1,\infty )$

$f\left(x\right)=x^2+1+\frac{1}{x^2+1}-1$
$x^2+1+\frac{1}{x^2+1}$ $\ge$ $2[\because AM\ge GM]$
$x^2+\frac{1}{x^2+1}\ge 1$
$\therefore$ $f\left(x\right)ϵ[1,\infty )$