Range of the function f ( x )= cot -1 - x + sin -1 x + cos -1 x where denotes fractional part function, is

Question

Range of the function f ( x )= cot -1 - x + sin -1 x + cos -1 x where denotes fractional part function, is (A) ((π/2), (3 π/2)) (B) ((π/2), π) (C) ((3 π/4), π] (D) [(3 π/4), π)

Tardigrade Admin · Accepted Answer

Θ f ( x )=(π/2)+ cot -1 - x Θ 0 ≤ - x <1 ⇒ (π/4)< cot -1 - x ≤ (π/2) ⇒ (3 π/4)< f ( x ) ≤ π

Q. Range of the function $f (x) = cot^{- 1} {- x} + sin^{- 1} {x} + cos^{- 1} {x}$ where \{\} denotes fractional part function, is

$(\frac{π}{2}, \frac{3 π}{2})$

$(\frac{π}{2}, π)$

$(\frac{3 π}{4}, π]$

$[\frac{3 π}{4}, π)$

$Θ f (x) = \frac{π}{2} + cot^{- 1} {- x}$
$Θ0 \leq {- x} < 1$
$\Rightarrow \frac{π}{4} < cot^{- 1} {- x} \leq \frac{π}{2}$
$\Rightarrow \frac{3 π}{4} < f (x) \leq π$

Q. Range of the function f(x)=cot−1{−x}+sin−1{x}+cos−1{x} where \{\} denotes fractional part function, is

(2π​,23π​)

(2π​,π)

(43π​,π]

[43π​,π)