Show that f(x)=1 1+x^2 is neither increasing nor decreasing on R … - Vidyadip

Question

Show that $f(x)=\frac{1}{1+x^2}$ is neither increasing nor decreasing on $R .$

✓

Answer

$f(x)=\frac{1}{1+x^2}$
Let $x _1> x _2$
$\Rightarrow x_1^2>x_2^2$
$\Rightarrow 1+x_1^2>1+x_2^2$
$\Rightarrow \frac{1}{1+x_1^2}<\frac{1}{1+x_2^2}$
$\Rightarrow f\left(x_1\right) < f\left(x_2\right)$
$f(x)$
is decreasing on $[0, \infty)$
Case $2$
$ \Rightarrow x_1^2 < x_2^2$
$\Rightarrow 1+x_1^2 < 1+x_2^2$
$\Rightarrow \frac{1}{1+x_1^2} > \frac{1}{1+x_2^2}$
$\Rightarrow f\left(x_1\right)>f\left(x_2\right) $
So, $f(x)$ is increasing on $[0, \infty]$
Thus, $f(x)$ is neither increasing nor decreasing on $R$

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