ILLUSTRATION 2.47
Find the range of f(x) = sin’x – 3 sinx + 2.
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Asked on December 27, 2019 by
Shilpa Kosvi
If [2cosx]+[sinx]=−3, then the range of the function, f(x)=sinx+
3
cosx in [0,2π] Where [ ] denotes the greatest integer function.
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ANSWER
Since, sinx,cosx ϵ[1,−1]
So, for equality to hold true in
[2cosx]+[sinx]=−3,
cosx ϵ[−1,
2
−1
) and sinx ϵ[−1,0) in x ϵ[0,2π]
For sinx ϵ[−1,0)
x ϵ[
3
2π
,
3
4π
]
[f(x)=
3
cosx+sinx[=2[
2
3
cosx+
2
1
sinx]
f(x)=2[sinxcos(π/3)+sin(π/3)cosx]=2sin(x+
3
π
)
for
π<x<40/3
π+
3
π
<x+
3
π
<4π+
3
π
3
4π
<x+
3
π
<
3
5π
sin(x+
3
π
) ϵ[−1,−
2
3
)
2sin(x+
3
π
) ϵ[−2,−
3
)
solution
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