已知函数y=1+(sinx)^2+根号3*cosx*sinx（1）求y的最值及对应x值（2）函数单调增区间

Question

已知函数y=1+(sinx)^2+根号3*cosx*sinx
（1）求y的最值及对应x值
（2）函数单调增区间

Answer 1

(1)
y=1+(sinx)^2+根号3*cosx*sinx
=1+(1-cos2x)/2+根号3/2*sin2x
=sin(2x-π/6)+3/2
所以最大值为5/2,此时,2x-π/6=2kπ+π/2,x=kπ+π/3(k∈Z)
最小值为1/2,此时,2x-π/6=2kπ-π/2,x=kπ-π/6(k∈Z)
(2)
单调递增时,2kπ-π/2<2x-π/6<2kπ+π/2,即kπ-π/6单调递减时,2kπ+π/2<2x-π/6<2kπ+3π/2,即kπ+π/3所以递增区间为(kπ-π/6,kπ+π/3)(k∈Z)
递减区间为(kπ+π/3,kπ+5π/6)(k∈Z)