求值域y=2sinx+cos^2x,x∈[π/6,2π/3)

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求值域y=2sinx+cos^2x,x∈[π/6,2π/3)

求值域y=2sinx+cos^2x,x∈[π/6,2π/3)
求值域y=2sinx+cos^2x,x∈[π/6,2π/3)

求值域y=2sinx+cos^2x,x∈[π/6,2π/3)
y=2sinx+cos^2x
=2sinx+1 - sin²x
=-(sinx-1)²+2
已知x∈[π/6,2π/3),那么:
sinx∈[1/2,1]
所以当sinx=1即x=π/2时,函数有最大值为2;当sinx=1/2即x=π/6时,函数有最小值为7/4.
即函数的值域为[7/4,2]

y=2sinx+cos^2x=2sinx+1-sin^2x=-(sinx-1)^2+2
[7/4,2]