Question

The function f(x) = cot⁻¹ x + x increases in the interval
(a) (1, ∞)
(b) (−1, ∞)
(c) (−∞, ∞)
(d) (0, ∞)

Answer 1

(c) (−∞, ∞)

$$\begin{gathered} f\left(x\right)={\cot }^{-1}x+x \\ f\text{'}\left(x\right)=\frac{-1}{1+x^2}+1 \\ =\frac{-1+1+x^2}{1+x^2} \\ =\frac{x^2}{1+x^2}\ge 0,\forall x\in R \\ \text{So,}\mathit{\text{f}}\text{(}\mathit{\text{x}})\text{is increasing on}\left(-\infty ,\infty \right). \end{gathered}$$