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y=x+｜sin2X｜如何求单调取间?
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y=x+｜sin2X｜如何求单调取间?
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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]
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