(2014•海淀区二模)已知函数f(x)=23sinxcosx-2sin2x+a,a∈R.
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(2014•海淀区二模)已知函数f(x)=2
3 |
(Ⅰ)∵f(x)=
3sin2x+cos2x+a-1
=2sin(2x+
π
6)+a-1,
∴T=
2π
2=π,
∴函数f(x)的最小正周期为π.
(Ⅱ)令f(x)=0,即2sin(2x+
π
6)+a-1=0,
则a=1-2sin(2x+
π
6),
∵-1≤sin(2x+
π
6)≤1,
∴-1≤1-2sin(2x+
π
6)≤3,
∴若f(x)有零点,则实数a的取值范围是[-1,3].
3sin2x+cos2x+a-1
=2sin(2x+
π
6)+a-1,
∴T=
2π
2=π,
∴函数f(x)的最小正周期为π.
(Ⅱ)令f(x)=0,即2sin(2x+
π
6)+a-1=0,
则a=1-2sin(2x+
π
6),
∵-1≤sin(2x+
π
6)≤1,
∴-1≤1-2sin(2x+
π
6)≤3,
∴若f(x)有零点,则实数a的取值范围是[-1,3].
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