若|x|≤π/4,求函数y=2sin|x+π/3|-2sinx的最值

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若|x|≤π/4,求函数y=2sin|x+π/3|-2sinx的最值

若|x|≤π/4,求函数y=2sin|x+π/3|-2sinx的最值
若|x|≤π/4,求函数y=2sin|x+π/3|-2sinx的最值

若|x|≤π/4,求函数y=2sin|x+π/3|-2sinx的最值
|x|≤π/4
-π/4≤x≤π/4
∴x+π/3>0
∴y=2sin|x+π/3|-2sinx
=2sin(x+π/3)-2sinx
=2(sinx*1/2+cosx*√3/2)-2sinx
=sinx+√3cosx-2sinx
=√3cosx-sinx
=2(cosx*√3/2-sinx*1/2)
=2cos(x+π/6)
-π/4≤x≤π/4
∴-π/12≤x+π/6≤5π/12
∴最小值
=2cos(π/4+π/6)
=2(√2/2*√3/2-√2/2*1/2)
=(√6-√2)/2
最大值
=2cos(-π/6+π/6)
=2*1
=2