Question

The function $f(x)=\frac{x}{1+x^2}$ decreases in the interval

• A. $(-\infty ,-1]\cup [1,\infty )$
• B. $(-1,1)$
• C. $(-\infty ,\infty )$
• D. $Noneoftheabove$

Answer 1

Answer: (A)

Given, $f(x)=\frac{x}{1+x^2}$
On differentiating w. r. t. x, we get
$f^{\prime }(x)=\frac{(1+x^2)(1)-x(2x)}{{(1+x^2)}^2}$
$=\frac{1-x^2}{{(1+x^2)}^2}$
For $f(x)$ to be decreasing
$f^{\prime }(x)\le 0$
$\Rightarrow$ $(1-x^2)\le 0$
$\Rightarrow$ $x^2\ge 1$
$\Rightarrow$ $x\in (-\infty ,-1]\cup [1,\infty )$