已知函数f(X)=4sin的2次方x+4根号3sinxcosx-3

f(x)=4(sinx)^2+4√3sinxcosx-3
=2(1-cos2x)+2√3sin2x-3
=2√3sin2x-2cos2x-1
=4(√3/2sin2x-1/2cos2x)-1
=4(sin2xcos(π/6)-cos2xsin(π/6))-1
=4sin(2x-π/6)-1
所以
1.f(x)的最小正周期是 2π/2=π
2.f(x)=4sin(2(x-π/12))-1
所以单调增区别是 [kπ-π/4+π/12,kπ+π/4+π/12]
即 [kπ-π/6,kπ+π/3] k∈Z
3.当x ∈[0,π/2]时
f(x)=4sin(2x-π/6))-1
显然,在 -π/6

(sinx)^2=1/2-cos(2x)/2
f(X)=4sin的2次方x+4根号3sinxcosx-3=
=(2-2cos(2x))+2sin(2x)-3=
=2(sin(2x)-cos(2x))-1=
=2sqrt(2)sin(2x-π/4)-1
1.f(X)的最小正周期为π
2.f(x)的单调增区间(2kπ-π/8,2kπ+7π/8)
3若.x属于{0,π/2}，f(x)的最大值为2sqrt(2)-1,最小值为-3