If y = 2x + ^ - 1 ,x + ( 1 + x^2 - x ), then y - Vidyadip

MCQ

If $y = 2x + {\cot ^{ - 1}}\,x + \log \left( {\sqrt {1 + {x^2}} - x} \right),$ then $y$

A
decrease on $\left( { - \infty ,\infty } \right)$
B
decreases on $\left[ {0,\infty } \right)$
C
Decreases on $\left[ {0,\infty } \right)$ and increase on $\left( { - \infty ,0} \right]$
✓
increases on $\left( { - \infty ,\infty } \right)$

✓

Answer

Correct option: D.

increases on $\left( { - \infty ,\infty } \right)$

d
$\frac{d y}{d x}=2-\frac{1}{1+x^{2}}-\frac{1}{\sqrt{1+x^{2}}} \geq 0 \quad \forall x \in R$

Since $\frac{1}{\left(1+x^{2}\right)}$ and $\frac{1}{\sqrt{1+x^{2}}}$ are less than or equal to $1$ for all $x$. So $f(x)$ increases on $(-\infty, \infty)$

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