The function f(x)= ^ -1 x+x increases in the interval: 1. (1, ) 2… - Vidyadip

Question

The function $\text{f}(\text{x})=\cot^{-1}\text{x}+\text{x}$ increases in the interval:

$(1,\infty)$
$(-1,\infty)$
$(-\infty,\infty)$
$(0,\infty)$

✓

Answer

$(-\infty,\infty)$

Solution:
$\text{f}(\text{x})=\cot^{-1}\text{x}+\text{x}$
$\text{f}'(\text{x})=\frac{-1}{1+\text{x}^2}+1$
f(x) is increasing,
$\Rightarrow\frac{-1}{1+\text{x}^2}+1>0$
$\Rightarrow\frac{\text{x}^2}{1+\text{x}^2}>0$
Hence, f(x) is increasing on $(-\infty,\infty).$

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