已知函数f(x)=1+sinxcosx,g(x)=cos²(x+π/12)(1)设x=x0是函数f(x)的图象的一条对称轴,求g(x0)的值.(2)求使函数h(x)=f(wx/2)=g(wx/2)(w>0)在区间[-2π/3,π/3]上是增函数的w的最大值.感激不尽!
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![已知函数f(x)=1+sinxcosx,g(x)=cos²(x+π/12)(1)设x=x0是函数f(x)的图象的一条对称轴,求g(x0)的值.(2)求使函数h(x)=f(wx/2)=g(wx/2)(w>0)在区间[-2π/3,π/3]上是增函数的w的最大值.感激不尽!](/uploads/image/z/2832311-47-1.jpg?t=%E5%B7%B2%E7%9F%A5%E5%87%BD%E6%95%B0f%28x%29%3D1%2Bsinxcosx%2Cg%28x%29%3Dcos%26sup2%3B%28x%2B%CF%80%2F12%29%281%29%E8%AE%BEx%3Dx0%E6%98%AF%E5%87%BD%E6%95%B0f%28x%29%E7%9A%84%E5%9B%BE%E8%B1%A1%E7%9A%84%E4%B8%80%E6%9D%A1%E5%AF%B9%E7%A7%B0%E8%BD%B4%2C%E6%B1%82g%28x0%29%E7%9A%84%E5%80%BC.%282%29%E6%B1%82%E4%BD%BF%E5%87%BD%E6%95%B0h%28x%29%3Df%28wx%2F2%29%3Dg%28wx%2F2%29%28w%3E0%29%E5%9C%A8%E5%8C%BA%E9%97%B4%5B-2%CF%80%2F3%2C%CF%80%2F3%5D%E4%B8%8A%E6%98%AF%E5%A2%9E%E5%87%BD%E6%95%B0%E7%9A%84w%E7%9A%84%E6%9C%80%E5%A4%A7%E5%80%BC.%E6%84%9F%E6%BF%80%E4%B8%8D%E5%B0%BD%21)
已知函数f(x)=1+sinxcosx,g(x)=cos²(x+π/12)(1)设x=x0是函数f(x)的图象的一条对称轴,求g(x0)的值.(2)求使函数h(x)=f(wx/2)=g(wx/2)(w>0)在区间[-2π/3,π/3]上是增函数的w的最大值.感激不尽!
已知函数f(x)=1+sinxcosx,g(x)=cos²(x+π/12)
(1)设x=x0是函数f(x)的图象的一条对称轴,求g(x0)的值.
(2)求使函数h(x)=f(wx/2)=g(wx/2)(w>0)在区间[-2π/3,π/3]上是增函数的w的最大值.
感激不尽!
已知函数f(x)=1+sinxcosx,g(x)=cos²(x+π/12)(1)设x=x0是函数f(x)的图象的一条对称轴,求g(x0)的值.(2)求使函数h(x)=f(wx/2)=g(wx/2)(w>0)在区间[-2π/3,π/3]上是增函数的w的最大值.感激不尽!
已知函数f(x)=1+sinxcosx,g(x)=[cos(x+(π/12))]^2
f(x)=1+sinxcosx=1+(1/2)sin2x
g(x)=[cos(x+(π/12)]^2=[cos(2x+(π/6))+1]/2
(1)设X=Xo是函数y=f(x)图像的一条对称轴,求g(x).
因为f(x)=1+(1/2)sin2x,对于正弦函数来说,当x=xo为对称轴时函数f(x)取得最大值或者最小值.即:sin2x=1,或者sin2x=-1
所以,2x=2xo=kπ+(π/2)(k∈Z)
那么,g(x)=[cos(2x+(π/6))+1]/2=[cos(kπ+(π/2)+(π/6))+1]/2
=[cos(kπ+(2π/3))+1]/2
当k为偶数时,g(x)=[cos(2π/3)+1]/2=[(-1/2)+1]/2=1/4
当k为奇数时,g(x)=[cos(5π/3)+1]/2=[(1/2)+1]/2=3/4
(2)求h(x)=f(wx/2)+g(wx/2)(w>0)在区间[-2π/3,π/3]上是增函数的w的最大值.
由前面知,f(x)=1+(1/2)sin2x,g(x)=[cos(2x+(π/6))+1]/2
所以,f(wx/2)=1+(1/2)sin(2*wx/2)=1+(1/2)sin(wz)
g(wx/2)=[cos(2*wx/2+(π/6))+1]/2=[cos(wx+(π/6))+1]/2
所以:f(wx/2)+g(wx/2)=1+(1/2)sin(wx)+(1/2)cos(wx+(π/6))+(1/2)
=(3/2)+(1/2)[sin(wx)+cos(wx+(π/6))]
=(3/2)+(1/2)[sin(wx)+cos(wx)*cos(π/6)-sin(wx)*sin(π/6)]
=(3/2)+(1/2)[sin(wx)+cos(wx)*(√3/2)]-sin(wx)*(1/2)]
=(3/2)+(1/2)[(1/2)sin(wx)+(√3/2)cos(wx)]
=(3/2)+(1/2)sin[(wx)+(π/3)]
=(3/2)+(1/2)sin[w(x+(π/3w))]
则其周期为T=2π/w
区间[-2π/3,π/3]的长度为(π/3)-(-2π/3)=π
要保证其在[-2π/3,π/3]上为增函数,则:
π/3w≥2π/3,且T/4=π/(2w)≥π
所以,w≤1/2
即,w的最大值为1/2