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6. Application of derivatives — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMaths6. Application of derivatives1 Mark
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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