证明|cosx-cosy|<|x-y|原题是小于等于。我不会打。 利用拉格朗日中值定理证明：对于任意实数x和y，有|cosx-...

\u8bc1\u660e\u4e0d\u7b49\u5f0fcosx-cosy\u7684\u7edd\u5bf9\u503c\u5c0f\u4e8e\u7b49\u4e8ex-y\u7684\u7edd\u5bf9\u503c\u63d0\u793a\u7528\u62c9\u683c\u6717\u65e5\u4e2d\u503c\u5b9a\u7406\u8bc1\u660e

cosx-cosy=-sin\u03b5(x-y),\u03b5\u4f4d\u4e8ex,y\u4e4b\u95f4
\u7531\u4e8e-sin\u03b5\u7684\u7edd\u5bf9\u503c\u4e00\u5b9a\u5c0f\u4e8e1\uff0c
\u6240\u4ee5cosx-cosy\u7684\u7edd\u5bf9\u503c\u4e00\u5b9a\u5c0f\u4e8ex-y

\u8bbef(x)=cosx \u5219f\u2018(x)=-sinx
f(x)\u5728[y\uff0cx]\u7b26\u5408\u62c9\u683c\u6717\u65e5\u4e2d\u503c\u5b9a\u7406\u6761\u4ef6
|cosx-cosy|/|x-y|=|-sin\u03be |<=1
\u2234 |cosx-cosy|<=|x-y|

根据和差化积公式，有：
|cosx-cosy|
=|-2sin((x+y)/2)*sin((x-y)/2)|
=2|sin((x+y)/2)*sin((x-y)/2)|
而 |sint|<=|t|
|sint|<=1
所以有：
2|sin((x+y)/2)*sin((x-y)/2)|<=2*1*|(x-y)/2|=|x-y|
所以有 |cosx-cosy|<=|x-y|

令x=y
则|cosx-cosy|=|x-y|，当x不等于y,且x是y的函数(cosx-cosy)'=sinydy/dx-sinx
|cosx-cosy|'=sinx-sinydy/dx=sinx(1-siny/sinxdy/dx)
|x-y|'=1-dy/dx,假设y=x,|cosx-cosy|'<=|x-y|',于是|cosx-cosy|<=|x-y|,所以在y=x这个函数下，|cosx-cosy|<=|x-y|

|cosx-cosy|<=|x-y|
|cosx-cosy|
=|2sin(x+y)/2sin(x-y)/2|
<=|2sin(x-y)/2|
<=|2(x-y)/2|
<=|x-y|

见参考资料

璇佹槑|cosx-cosy|
绛旓細|cosx-cosy| =|-2sin((x+y)/2)*sin((x-y)/2)| =2|sin((x+y)/2)*sin((x-y)/2)| 鑰 |sint|

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绛旓細2|sin((x+y)/2)*sin((x-y)/2)|<=2*1*|(x-y)/2|=|x-y| 鎵浠ユ湁 |cosx-cosy|<=|x-y|

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绛旓細瀹规槗銆傜敤宸寲绉叕寮cosx-cosy=-2sin(x+y)/2sin(x-y)/2 鎵浠ュ乏杈癸紳2|sin(x+y)/2|*|sin(x-y)/2|<=2|sin(x-y)/2|<=2|(x-y)/2|=|x-y|

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鎴浘涓婄殑涓夎鍑芥暟鐨勬棤绌蜂箻绉庝箞璇佹槑鍟
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鐢ㄥ畾涔璇佹槑浣欏鸡鍑芥暟閫掑噺
绛旓細cosx-cosy=-2sin[(x+y)/2]sin[(x-y)/2] [cos(x+dx)-cosx]/dx = -2sin(x+dx/2)*sin(dx/2)/dx= -sin(x+dx/2) (sin(dx/2)/(dx/2)) dx->0 sin(dx/2)/(dx/2) ->1 sin(x+dx/2) -> sinx銆

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