已知函数f(x)=2cos^2x+6sinxcosx+4cos(x+45度)cos(x-45度）

f(x)=2cos^2x+6sinxcosx+4cos(x+π/4)cos(x-π/4）
=2cos^2x+6sinxcosx+4(√2/2cosx+√2/2sinx)(√2/2cosx-√2/2sinx)
=2cos^2x+6sinxcosx+2cos^2x-2sin^x
=6cos^2x+3sin2x-2
=3(2cos^2x-1)+3sin2x+1
=3cos2x+3sin2x+1
=3√2sin(2x+π/4)+1
1.T=2π/2=π
2.x∈[0,π/4]
2x∈[0,π/2]
2x+π/4∈[π/4,3π/4]
f(x)的最小值=3√2*√2/2+1=4
f(x)的最大值=3√2*1+1=3√2+1
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(1)f(x)=1+cos2x+3sin2x+4[√2/2(cosx-sinx)][√2/2(cosx+sinx)]
=1+cos2x+3sin2x+2cos2x
=1+3√2(√2/2cos2x+√2/2sin2x)=1+3√2sin(2x+π/4)
f(x)的最小正周期为T=2π/2=π
(2)x∈[0,π/4]
2x+π/4∈[π/4,3π/4]