Question

The function $y=x/(1+x^2)$ decreases in the interval

• A. $(-1,1)$
• B. $(1,\infty )$
• C. $(-\infty ,-1)$
• D. $(-\infty ,\infty )$

Answer 1

Answer: (B), (C)

$(1,\infty )$
$(-\infty ,-1)$

$y=x/(1+x^2)$
$y^{\prime }=\frac{(1+x^2)-2x\cdot x}{{(1+x^2)}^2}=\frac{1-x^2}{{(1+x^2)}^2}$
Since ${(1+x^2)}^2>0$ for all $x$
Therefore $y^{\prime }<0$ for $1-x^2<0$
$\Rightarrow$ $x>1$ or $x<-1$
Therefore, $x\in (-\infty ,-1)$ or $x\in (1,\infty )$.
Thus $x/(1+x^2)$ decreases in $(-\infty ,-1]$ or $[1,\infty )$.
Ans: B,C