求函数y=sin(x+π/6)+cosx,(0≤x≤π)的最大值和最小值
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求函数y=sin(x+π/6)+cosx,(0≤x≤π)的最大值和最小值
求函数y=sin(x+π/6)+cosx,(0≤x≤π)的最大值和最小值
求函数y=sin(x+π/6)+cosx,(0≤x≤π)的最大值和最小值
y=sin(x+π/6)+cosx
=sinxcos(π/6)+cosxsin(π/6)+cosx
=(√3/2)*sinx+(3/2)cosx
=√3*[(1/2)*sinx+(√3/2)cosx]
=√3*sin(x+π/3)
因为0≤x≤π,即π/3≤x+π/3≤4π/3
所以-√3/2≤sin(x+π/3)≤1
则当x+π/3=π/2,即x=π/6时,sin(x+π/3)=1,函数y有最大值√3;
当x+π/3=4π/3,即x=π时,sin(x+π/3)=-√3/2,函数y有最小值-3/2.
y=sin(x+π/6)+cosx=sinxcosπ/6+cosxsinπ/6+cosx=(√3/2)sinx+(1/2)cosx+cosx=(√3/2)sinx+(3/2)cosx=√3[(1/2)sinx+(√3/2)cosx]=√3(sinxcosπ/3+cosxsinπ/3)=√3sin(x+π/3).
当0≤x≤π时,π/3≤x+π/3≤4π/3,sin(4π/3)≤sin(x+...
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y=sin(x+π/6)+cosx=sinxcosπ/6+cosxsinπ/6+cosx=(√3/2)sinx+(1/2)cosx+cosx=(√3/2)sinx+(3/2)cosx=√3[(1/2)sinx+(√3/2)cosx]=√3(sinxcosπ/3+cosxsinπ/3)=√3sin(x+π/3).
当0≤x≤π时,π/3≤x+π/3≤4π/3,sin(4π/3)≤sin(x+π/3)≤sin(π/2),即-√3/2≤sin(x+π/3)≤1,所以:-3/2≤√3sin(x+π/3)≤√3.也即-3/2≤y≤√3。所以最大值和最小值分别是:√3和-3/2,分别当x+π/3=π/2即x=π/6和x+π/3=4π/3即x=π时取得。
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化简结果为sin(x+£/3)最大值为1最小值为~1