已知函数f(x)=sin^3xcosx+cos^3xsinx+√3sin^2x求函数的单调减区间求y=(x)(0≤x≤ π)的值域
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![已知函数f(x)=sin^3xcosx+cos^3xsinx+√3sin^2x求函数的单调减区间求y=(x)(0≤x≤ π)的值域](/uploads/image/z/3711655-55-5.jpg?t=%E5%B7%B2%E7%9F%A5%E5%87%BD%E6%95%B0f%28x%29%3Dsin%5E3xcosx%2Bcos%5E3xsinx%2B%E2%88%9A3sin%5E2x%E6%B1%82%E5%87%BD%E6%95%B0%E7%9A%84%E5%8D%95%E8%B0%83%E5%87%8F%E5%8C%BA%E9%97%B4%E6%B1%82y%3D%28x%29%280%E2%89%A4x%E2%89%A4+%CF%80%EF%BC%89%E7%9A%84%E5%80%BC%E5%9F%9F)
已知函数f(x)=sin^3xcosx+cos^3xsinx+√3sin^2x求函数的单调减区间求y=(x)(0≤x≤ π)的值域
已知函数f(x)=sin^3xcosx+cos^3xsinx+√3sin^2x求函数的单调减区间求y=(x)(0≤x≤ π
)的值域
已知函数f(x)=sin^3xcosx+cos^3xsinx+√3sin^2x求函数的单调减区间求y=(x)(0≤x≤ π)的值域
f(x)=sin³xcosx+cos³xsinx+√3 sin²x
=sinxcosx(sin²x+cos²x)+√3 (1-cos2x)/2
=½ sin2x - √3 /2 cos2x + √3 /2
=sin(2x-π/3) + √3 /2
f(x) 的递减区间是 2x-π/3∈[π/2+kπ,3π/2+kπ] (k∈Z)
即 x∈[5π/12+kπ/2,11π/12+kπ/2] (k∈Z)
在 0≤x≤π 内的单调递减区间是 x∈[5π/12,11π/12]
值域是 [√3 /2 - 1,√3 /2 + 1]
已知函数f(x)=sin³xcosx+cos³xsinx+(√3)sin²x,求函数的单调减区间,求y=f(x)(0≤x≤ π )的值域
f(x)=sinxcosx(sin²x+cos²x)+(√3/2)(1-cos2x)=(1/2)sin2x-(√3/2)cos2x+(√3)/2
=sin2xcos(π/3)-cos2xsin(π...
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已知函数f(x)=sin³xcosx+cos³xsinx+(√3)sin²x,求函数的单调减区间,求y=f(x)(0≤x≤ π )的值域
f(x)=sinxcosx(sin²x+cos²x)+(√3/2)(1-cos2x)=(1/2)sin2x-(√3/2)cos2x+(√3)/2
=sin2xcos(π/3)-cos2xsin(π/3)+(√3)/2=sin(2x-π/3)+(√3)/2
单调递减区间:由π/2+2kπ≦2x-π/3≦3π/2+2kπ,得5π/6+2kπ≦2x≦11π/6+2kπ,故单调递减区间为:5π/12+kπ≦x≦11π/12+kπ,k∈Z.
当0≦x≦ π时,[(√3)-2]/2≦f(x)≦[(√3+2)/2].
其中,f(11π/12)=sin(11π/6-π/3)+(√3)/2=sin(π+π/2)+(√3)/2=-1+(√3)/2=[(√3)-2]/2
f(5π/12)=sin(5π/6-π/3)+(√3)/2=sin(π/2)+(√3)/2=[(√3)+2]/2
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