已知函数f(x)=sin^2x+4sinxcosx-3cos^2x,x属于R.求函数的最小正周期

f(x)=1-cos²2x+4sinxcosx-3cos²2x
=2sin2x-1-4cos²2x
=2sin2x-2(2cos²2x-1)-3
=2sin2x-2cos2x-3
=2√2sin(2x-π/4)-3
∵f(x)=2√2sin(2x-π/4)-3
∴T=2π/2=π
f(x)=2√2sin(2x-π/4)-3
单调增区间为[kπ-π/8,kπ+3π/8]k∈Z
单调减区间为[kπ+3π/8,kπ+7π/8]k∈Z
当x∈（-π/4,π/3）当x=-π/8时 f(x)min=-2√2-3
当x=π/3时 f(x)max=-2+√3
∴f(x)∈[-2√2-3,-2+√3)