已知函数f(x)=cos的4次方x+2sinxcosx--sin的4次方x 已知函数f(x)=cosx的4次方+2sinxcosx-si...
\u5df2\u77e5\u51fd\u6570f(x)=cos\u56db\u6b21\u65b9x-2sinxcosx-sin\u56db\u6b21\u65b9xf(x)=(cosx)^4-2sinxcosx-(sinx)^4
=[(cosx)^2+(sinx)^2][(cosx)^2-(sinx)^2]-2sinxcosx
=(cosx)^2-(sinx)^2-2sinxcosx
=cos(2x)-sin(2x)
= \u221a2*[cos(2x)*\u221a2/2-sin(2x)*\u221a2/2]
= \u221a2*[cos(2x)cos(\u03c0/4)-sin(2x)sin(\u03c0/4)]
= \u221a2cos(2x+\u03c0/4) \u3002
\uff082\uff09\u56e0\u4e3a 0<=x<=\u03c0/2\uff0c\u56e0\u6b64 \u03c0/4<=2x+\u03c0/4<=5\u03c0/4 \uff0c
\u6240\u4ee5\u5f53 2x+\u03c0/4= \u03c0 \u5373 x=3\u03c0/8 \u65f6 f(x) \u6700\u5c0f\u503c\u4e3a f(3\u03c0/8)= -\u221a2 \u3002
f(x)=(cos^2x +sin^2 x)(cos^2 x-sin^2 x)+sin2x
=cos2x+sin2x
=\u221a2sin(2x+\u03c0/4)
\u6700\u5c0f\u6b63\u5468\u671f\u4e3a\u03c0
\u5355\u8c03\u589e\u533a\u95f4\u4e3a2k\u03c0-\u03c0/2=<2x+\u03c0/4<=2k\u03c0+\u03c0/2, \u5373k\u03c0-3\u03c0/8=<x<=k\u03c0+\u03c0/8
\u8fd9\u91cck\u4e3a\u4efb\u610f\u6574\u6570\u3002
=(cos的2次方x-sin的2次方x)(cos的2次方x+sin的2次方x)+sin2x
=cos的2次方x-sin的2次方x+sin2x
∵cos2x=cos的2次方x-sin的2次方x
∴f(x)=cos2x+sin2x=√2sin(2x+π/4)
∵x∈[0,π/2]
∴π/4 ≤2x+π/4≤ 5π/4
f(x)的最小值=f(5π/4)=-√2/2
集合为﹛xⅠx= -π/8+kπ/2,k∈Z﹜
有点难看，不过应该能帮的到你
f(x)=(cos^2x+sin^2x)(cos^2x-sin^2x)+sin2x
=cos2x+sin2x=根号2sin（2x+45）
所以最小值为-根号2
最小值时的集合为 -90+360k《2x+45《90+360k
可得 -67.5+180k《x《22.5+180k
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