FunctionHard
Question
If [2 cos x] + [sin x] = - 3, then the range of the function, f(x) = sin x + √3 cos x in [0, 2 π] is:
(where [. ] denotes greatest integer function)
(where [. ] denotes greatest integer function)
Options
A.[- 2, -1)
B.(- 2, - 1]
C.(- 2, -1)
D.[-2, -√3)
Solution
[2 cos x] + [sin x] = - 3
-2 ≤ 2 cos x ≤ 2 ⇒ [2 cos x] = 2, 1, - 1, - 2
- 1 ≤ sin x ≤ 1 ⇒ [sin x] = 1, 0, - 1
Equation holds true for [2 cos x] = - 2 and [sin x] = -1
⇒ - 1 ≤ cos x < -
and - 1 ≤ sin x < 0
⇒ x ∈
and x ∈ (π, 2π)
⇒ x ∈
f(x) = sin x + √3 cos x
f(x) = 2 sin
⇒ - 1 ≤ sin
⇒ - 2 ≤ 2 sin
< - √3
Hence range is [-2, -√3]
-2 ≤ 2 cos x ≤ 2 ⇒ [2 cos x] = 2, 1, - 1, - 2
- 1 ≤ sin x ≤ 1 ⇒ [sin x] = 1, 0, - 1
Equation holds true for [2 cos x] = - 2 and [sin x] = -1
⇒ - 1 ≤ cos x < -
and - 1 ≤ sin x < 0⇒ x ∈
and x ∈ (π, 2π)⇒ x ∈
f(x) = sin x + √3 cos x
f(x) = 2 sin
⇒ - 1 ≤ sin
⇒ - 2 ≤ 2 sin
< - √3Hence range is [-2, -√3]
Create a free account to view solution
View Solution FreeMore Function Questions
If f(x) is defined on (0, 1) then the domain of definition of f(ex) + f(ln | x | ) is -...The range of the function f(x) = ex-e-x, is -...Domain of the function log |x2 − 9| is-...If f : (e, ∞) → R & f(x) f(x) = log [log(log x)], then is -...If f(x) = cot-1x : R+ → and g(x) = 2x - x2 : R → R. Then the range of the function f(g(x)) wherever define i...