作业帮已知w>0,a向量=(2sinwx+coswx,2sinwx-coswx)b向量=(sinwx,coswx),f(x)=a向量*b向量,f(x)=a向量*b向量,f(x)图像上相邻的两条对称轴的距离为π/2.求w的值求函数f(x)在[0,π/2]上的单调区间及最值
来源：学生作业帮助网 编辑：作业帮 时间：2024/05/03 23:05:31
![已知w>0,a向量=(2sinwx+coswx,2sinwx-coswx)b向量=(sinwx,coswx),f(x)=a向量*b向量,f(x)=a向量*b向量,f(x)图像上相邻的两条对称轴的距离为π/2.求w的值求函数f(x)在[0,π/2]上的单调区间及最值](/uploads/image/z/6931812-12-2.jpg?t=%E5%B7%B2%E7%9F%A5w%3E0%2Ca%E5%90%91%E9%87%8F%3D%282sinwx%2Bcoswx%2C2sinwx-coswx%29b%E5%90%91%E9%87%8F%3D%28sinwx%2Ccoswx%29%2Cf%28x%29%3Da%E5%90%91%E9%87%8F%2Ab%E5%90%91%E9%87%8F%2Cf%28x%29%3Da%E5%90%91%E9%87%8F%2Ab%E5%90%91%E9%87%8F%2Cf%28x%29%E5%9B%BE%E5%83%8F%E4%B8%8A%E7%9B%B8%E9%82%BB%E7%9A%84%E4%B8%A4%E6%9D%A1%E5%AF%B9%E7%A7%B0%E8%BD%B4%E7%9A%84%E8%B7%9D%E7%A6%BB%E4%B8%BA%CF%80%2F2.%E6%B1%82w%E7%9A%84%E5%80%BC%E6%B1%82%E5%87%BD%E6%95%B0f%28x%29%E5%9C%A8%5B0%2C%CF%80%2F2%5D%E4%B8%8A%E7%9A%84%E5%8D%95%E8%B0%83%E5%8C%BA%E9%97%B4%E5%8F%8A%E6%9C%80%E5%80%BC)
已知w>0,a向量=(2sinwx+coswx,2sinwx-coswx)b向量=(sinwx,coswx),f(x)=a向量*b向量,f(x)=a向量*b向量,f(x)图像上相邻的两条对称轴的距离为π/2.求w的值求函数f(x)在[0,π/2]上的单调区间及最值
已知w>0,a向量=(2sinwx+coswx,2sinwx-coswx)b向量=(sinwx,coswx),f(x)=a向量*b向量,
f(x)=a向量*b向量,f(x)图像上相邻的两条对称轴的距离为π/2.
求w的值
求函数f(x)在[0,π/2]上的单调区间及最值
已知w>0,a向量=(2sinwx+coswx,2sinwx-coswx)b向量=(sinwx,coswx),f(x)=a向量*b向量,f(x)=a向量*b向量,f(x)图像上相邻的两条对称轴的距离为π/2.求w的值求函数f(x)在[0,π/2]上的单调区间及最值
1
f(x)=a·b=(2sin(wx)+cos(wx),2sin(wx)-cos(wx))·(sin(wx),cos(wx))
=(2sin(wx)+cos(wx))sin(wx)+(2sin(wx)-cos(wx))cos(wx)
=2sin(wx)^2+sin(wx)cos(wx)+2sin(wx)cos(wx)-cos(wx)^2
=3sin(2wx)/2+(1-cos(2wx))-(1+cos(2wx))/2
=3sin(2wx)/2-3cos(2wx)/2+1/2
=(3√2/2)sin(2wx-π/4)+1/2
f(x)图像相邻的两条对称轴的距离为π/2
即f(x)的最小正周期：T=π
即：2π/(2w)=π
即：w=1
即：f(x)=(3√2/2)sin(2x-π/4)+1/2
2
增区间：2x-π/4∈[2kπ-π/2,2kπ+π/2]
即：x∈[kπ-π/8,kπ+3π/8],k∈Z
减区间：2x-π/4∈[2kπ+π/2,2kπ+3π/2]
即：x∈[kπ+3π/8,kπ+7π/8],k∈Z
x∈[0,π/2],故当k=0时,增区间：x∈[0,3π/8]
减区间：x∈[3π/8,π/2]
x∈[0,π/2],2x-π/4∈[-π/4,3π/4]
sin(2x-π/4)∈[-√2/2,1]
故：(3√2/2)sin(2x-π/4)+1/2∈[-1,(3√2+1)/2]
即：fmax=(3√2+1)/2
fmin=-1