1
向量a=(2根号3sinx,cos^2x),b=(cosx,2),
f(x)=a●b
=2√3sinxcosx+2cos²x
=√3sin2x+cos2x+1
=2(√3/2sin2x+1/2cos2x)+1
=2sin(2x+π/6)+1
由2kπ+π/2≤2x+π/6≤2kπ+3π/2,k∈Z
得kπ+π/6≤x≤2kπ+2π/3,k∈Z
∴函数f(x)的单调递减区间为
[kπ+π/6,2kπ+2π/3],k∈Z
2
将函数y=f(x)图像向左平移π/12个单位
得到y=2sin[2(x+π/12)+π/6]+1=2sin(2x+π/3)+1
图像,将所得图像上各点的横坐标缩短为
原来的1/2倍,纵坐标不变,得到函数
g(x)=2sin(4x+π/3)图像
∵x∈[0,π/4]
∴4x∈[0,π]
∴4x+π/3∈[π/3,4π/3]
∴4x+π/3=π/2时,g(x)max=3
4x+π/3=4π/3时,g(x)min=1-√3
∴g(x)值域为[1-√3,3]