y=|sin2x|con2x(x属于[0,π]) 求函数此时在定义域内的极大值和函数与x轴围成的面积 求详细讲解
\u5df2\u77e5\u51fd\u6570f(x)=cos2x/sin(x+\u03c0/4\uff09\uff0c\u6c42\u51fd\u6570f(x)\u7684\u5b9a\u4e49\u57df \u6c42\uff1a\u82e5f(x)=4/3,sin2x\u7684\u503c1\uff1a\u7531\u9898\u610f\u53ef\u77e5\uff0cf\uff08x\uff09\u7684\u5b9a\u4e49\u57df\u4e3a sin(x+\u03c0/4)\u22600 \u5f97\u51fax\u2260\uff08-\u03c0/4\uff09+2k\u03c0
2\uff1a\u7531\u9898\u610f\u53ef\u77e5\uff1a3cos2x=4sin(x+\u03c0/4\uff09\u21923cos²x-3sin²x=2*\u6839\u53f72\uff08sinx+cosx\uff09\u2192
3\uff08cosx+sinx\uff09\uff08cosx-sinx\uff09=2*\u6839\u53f72\uff08sinx+cosx\uff09 \u5219 cosx-sinx=\uff082*\u6839\u53f72\uff09/3
(cosx-sinx)²=8/9 \u21921-2sinxcosx=8/9 2sinxcosx=1/9=sin2x\u3002
2\u662f\u5e73\u65b9 \u7684\u8bdd \u90a3\u5c31\u662f \u8d1f\u65e0\u7a77 \u5230 \u6b63\u65e0\u7a77\uff0c
\u56e0\u4e3a x \u5728 \u8d1f\u65e0\u7a77 \u5230 \u6b63\u65e0\u7a77 sin^2(x) \u90fd\u5728 [0\uff0c1] \u5728 [0,4] \u4ee5\u5185\u3002
当x∈[0,π/2]时，2x∈[0,π],sin2x≥0
y=sin2xcos2x=1/2sin4x
y'=1/2cos4x*4=2cos4x
y'=0==>4x=π/2或4x=3π/2
==> x=π/8或x=3π/8
x∈(0,π/8),y'>0,函数递增
x∈(π/8,3π/8),y'<0，函数递减
x∈(3π/8,π/2)时，y>0，函数第增
∴x=π/8为函数极大值点
极大值为1/2
当x∈(π/2,π]时，2x∈(π,2π],sin2x≤0
y=-sin2xcos2x=-1/2sin4x
y'=1/2cos4x*4=2cos4x
y'=0==>4x=5π/2或4x=7π/2
==> x=5π/8或x=7π/8
x∈(π/2,5π/8),y'<0,函数递减
x∈(5π/8,7π/8),y'>0，函数递增
x∈(7π/8,π)时，y<0，函数第减
∴x=7π/8为函数极大值点
极大值为1/2
2
x∈[0,π/4]时，y=1/2sin4x≥0
在区间[0,π/4]上，函数与x轴围成的面积
S=ʃ(0-->π/4)1/2sin4xdx
=1/8ʃ(0-->π/4)(-cos4x)dx
=1/8*(cos0-cosπ)
=1/4
∴函数与x轴围成的面积为4S=1
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