y=x+|sin2x|等价于
y=x+sin2x kπ≤x≤kπ+π/2
y=x-sin2x kπ+π/2≤x≤(k+1)π
对该曲线方程求导,得
y'=1+2cos2x kπ≤x≤kπ+π/2
y'=1-2cos2x kπ+π/2≤x≤(k+1)π
当y'>0时,该函数单调增,即
1+2cos2x>0 kπ≤x≤kπ+π/2
1-2cos2x>0 kπ+π/2≤x≤(k+1)π
即±cos2x>-1/2
在kπ≤x≤kπ+π/2时,当且仅当kπ≤x≤kπ+π/3时,
cos2x>cos2π/3,即cos2x>-1/2
kπ+π/2≤x≤(k+1)π时,当且仅当kπ+π/2≤x≤kπ+5π/6时,
cos2x-1/2,
因此y=x+|sin2x|单调增区间是[kπ,kπ+π/3]U[kπ+π/2,kπ+5π/6],
单调减区间是[kπ+π/3,kπ+π/2]U[kπ+5π/6,kπ+π]