已知函数f(x)=4sinxsin^2(∏/4+x/2)+cos2x,x∈[π/6,4π/3],求f(x)最大值与最小值,并求对应x的值
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![已知函数f(x)=4sinxsin^2(∏/4+x/2)+cos2x,x∈[π/6,4π/3],求f(x)最大值与最小值,并求对应x的值](/uploads/image/z/3794186-2-6.jpg?t=%E5%B7%B2%E7%9F%A5%E5%87%BD%E6%95%B0f%28x%29%3D4sinxsin%5E2%28%E2%88%8F%2F4%2Bx%2F2%29%2Bcos2x%2Cx%E2%88%88%5B%CF%80%2F6%2C4%CF%80%2F3%5D%2C%E6%B1%82f%28x%29%E6%9C%80%E5%A4%A7%E5%80%BC%E4%B8%8E%E6%9C%80%E5%B0%8F%E5%80%BC%2C%E5%B9%B6%E6%B1%82%E5%AF%B9%E5%BA%94x%E7%9A%84%E5%80%BC)
已知函数f(x)=4sinxsin^2(∏/4+x/2)+cos2x,x∈[π/6,4π/3],求f(x)最大值与最小值,并求对应x的值
已知函数f(x)=4sinxsin^2(∏/4+x/2)+cos2x,x∈[π/6,4π/3],求f(x)最大值与最小值,并求对应x的值
已知函数f(x)=4sinxsin^2(∏/4+x/2)+cos2x,x∈[π/6,4π/3],求f(x)最大值与最小值,并求对应x的值
∵f(x)=4sinxsin²(π/4+x/2)+cos(2x)
=2sinx[2sin²(π/4+x/2)]+cos(2x)
=2sinx[1-cos(2(π/4+x/2))]+cos(2x)
=2sinx[1-cos(π/2+x)]+cos(2x)
=2sinx(1+sinx)+1-2sin²x
=2sinx+1
∴令f '(x)=2cosx=0,则 x=π/2 (∵x∈[π/6,4π/3])
∵f(π/6)=2sin(π/6)+1=2
f(π/2)=2sin(π/2)+1=3
f(4π/3)=2sin(4π/3)+1=1-√3
∴f(x)最大值是f(π/2)=3,最小值是f(4π/3)=1-√3.