【正文】 ragraphFormatLineSpacingLinesToPointselectionParagraaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaphFormatLineSpacingLinesToPointsSelectionParagraphFormatLineSpacingLinesTSelectionParbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbagraphFoLineSpacingLinesToPointsSelectionParagraphFormatLineSpacingLinesToPointse11111111111111111111111111111111lectionParagraphFormatLineSpacingLinesToPointsSelectionParagraphFormatLineSpacingLinesToPoctionParagraphFormatLineSpaci2222222222222222222222ngLinesToPoints2SelectionParagraphFormatLineSpacingLinesToPointsSelectionParagraphFormatLineSpacingLinesToPointselectionParagraphFccccccccccccccccccccccccccccccccccccccccccccccccccccccccormatLineSpacingLinesToPointsSelectionParagraphFormatLineSpacingLinesToPoctionParagraSelec SelectionParagraphFormatLineSpacingLinesToPointsSelectionParagraphFormatLineSpacingLinesToPointselectionParagraaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaphFormatLineSpacingLinesToPointsSelectionParagraphFormatLineSpacingLinesTSelectionParbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbagraphFoLineSpacingLinesToPointsSelectionParagraphFormatLineSpacingLinesToPointse11111111111111111111111111111111lectionParagraphFormatLineSpacingLinesToPointsSelectionParagraphFormatLineSpacingLinesToPoctionParagraphFormatLineSpaci2222222222222222222222ngLinesToPoints2SelectionParagraphFormatLineSpacingLinesToPointsSelectionParagraphFormatLineSpacingLinesToPointselectionParagraphFccccccccccccccccccccccccccccccccccccccccccccccccccccccccormatLineSpacingLinesToPointsSelectionParagraphFormatLineSpacingLinesToPoctionParagraSelec ,1}2 1,1{m a x)}({m a x 1]1,0[]1,0[ ?? ??? pxx xf ?? )}({min]1,0[ xfx ,2 1}2 1,1{m in 11]1,0[ ??? ? ppx 因此對(duì)一切 1],1,0[ ?? px 時(shí)，有 ,1)(2 1 1 ??? xfp 所以原不等式得證 . （ 1）所設(shè)函數(shù) )(xf 在某閉區(qū)間上連續(xù)，開(kāi)區(qū)間內(nèi)可導(dǎo)，但在所討論的區(qū)間上不是單調(diào)函數(shù)時(shí)；（ 2）只能證不嚴(yán)格的不等式而不能證出嚴(yán)格的不等式 . 四、用拉格朗日中值定理證明不等式法 －拉格朗日中值定理 拉格朗日中值定理：若函數(shù) )(xf 滿足下列條件：（ I） )(xf 在閉區(qū)間 ],[ ba 上連續(xù)；（ⅱ）)(xf 在開(kāi)區(qū)間 ),( ba 內(nèi)可導(dǎo)，則在 ),( ba 內(nèi)至少存在一點(diǎn) ? ，使得 ab afbff ???? )()()(? . 拉格朗日中值定理反映了函數(shù)或函數(shù)增量和可導(dǎo)函數(shù)的一階導(dǎo)數(shù)符號(hào)之間的關(guān)系 . ①輔助函數(shù) )(xf ，并確定 )(xf 施用拉格朗日中值定理的區(qū)間 ],[ ba ； ②對(duì) )(xf 在 ],[ ba 上施用拉格朗日中值定理； ③利用 ? 與 ba, 的關(guān)系，對(duì)拉格朗日公式進(jìn)行加強(qiáng)不等式 . SelectionParagraphFormatLineSpacingLinesToPointsSelectionParagraphFormatLineSpacingLinesToPointselectionParagraaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaphFormatLineSpacingLinesToPointsSelectionParagraphFormatLineSpacingLinesTSelectionParbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbagraphFoLineSpacingLinesToPointsSelectionParagraphFormatLineSpacingLinesToPointse11111111111111111111111111111111lectionParagraphFormatLineSpacingLinesToPointsSelectionParagraphFormatLineSpacingLinesToPoctionParagraphFormatLineSpaci2222222222222222222222ngLinesToPoints2SelectionParagraphFormatLineSpacingLinesToPointsSelectionParagraphFormatLineSpacingLinesToPointselectionParagraphFccccccccccccccccccccccccccccccccccccccccccccccccccccccccormatLineSpacingLinesToPointsSelectionParagraphFormatLineSpacingLinesToPoctionParagraSelec SelectionParagraphFormatLineSpacingLinesToPointsSelectionParagraphFormatLineSpacingLinesToPointselectionParagraaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaphFormatLineSpacingLinesToPointsSelectionParagraphFormatLineSpacingLinesTSelectionParbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbagraphFoLineSpacingLinesToPointsSelectionParagraphFormatLineSpacingLinesToPointse11111111111111111111111111111111lectionParagraphFormatLineSpacingLinesToPointsSelectionParagraphFormatLineSpacingLinesToPoctionParagraphFormatLineSpaci2222222222222222222222ngLinesToPoints2SelectionParagraphFormatLineSpacingLinesToPointsSelectionParagraphFormatLineSpacingLinesToPointselectionParagraphFccccccccccccccccccccccccccccccccccccccccccccccccccccccccormatLineSpacingLinesToPointsSelectionParagraphFormatLineSpacingLinesToPoctionParagraSelec 例 8：證明：當(dāng) xxxxx ????? )1ln(1,0 . 分析：所證不等式中的函數(shù) )1ln( x? 的導(dǎo)數(shù)為 x?11 ，即所證不等式中含有函數(shù)及其導(dǎo)數(shù)，因而可用拉格朗日中值定理試之 .由于 01ln? ，因此可構(gòu)造函數(shù)的改變量 1ln)1ln( ?? x ，則相應(yīng)自變量的改變量為 x ，原不等式等價(jià)于： 11)1( 11)1ln(1 1 ??? ???? x nxx，由不等式中間部分的形式可知，可利用拉格朗日中值定理去證明 . 證明：構(gòu)造函數(shù) ttf ln)( ? ，因 )(tf 在 )0](1,1[ ?? xx 上連續(xù)，在 )1,1( x? 上可導(dǎo)，)(tf 在 )0](1,1[ ?? xx 上 滿 足 拉 格 朗 日 條 件 ， 于 是 存 在 )1,1( x??? ，使 ?? 1)(1)1( )1()1( ????? ?? fx fxf，因 111 1),1l n(1ln)1l n()1()1( ?????????? ?xxxfxf ，所以1)1l n(1 ???? xxx . 即 )0(,)1ln(1 ????? xxxxx . 當(dāng)所證的不等式中含有函數(shù)值與 一階導(dǎo)數(shù)，或函數(shù)增量與一階導(dǎo)數(shù)時(shí)，可用拉格朗日中值定理來(lái)證明 . 五、用柯西中值定理證明不等式法 －柯西中值定理 SelectionParagraphFormatLineSpacingLinesToPointsSelectionParagraphFormatLineSpacingLinesToPointselectionParagraaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaphFormatLineSpacingLinesToPointsSelectionParagraphFormatLineSpacingLinesTSelectionParbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbagraphFoLineSpacingLinesToPointsSelectionParagraphFormatLineSpacingLinesToPointse11111111111111111111111111111111lectionParagraphFormatLineSpacingLinesToPointsSelectionParagraphFormatLineSpacingLinesToPoctionParagraphFormatLineSpaci2222222222222222222222ngLinesToPoints2SelectionParagraphFormatLineSpacingLinesToPointsSelectionParagraphFormatLineSpacingLinesToPointselectionParagraphFccccccccccccccccccccccccccccccccccccccccccccccccccccccccormatLineSpacingLinesToPointsSelectionParagraphFormatLineSpacingLinesToPoctionParagraSelec SelectionParagraphFormatLineSpacingLinesToPointsSelectionParagraphFormatLineSpacingLinesToPointselectionParagraaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaphFormatLineSpacingLinesToPointsSelectionParagraphFormatLineSpacingLinesTSelectionParbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbagraphFoLineSpacingLinesToPointsSelectionParagraphFormatLineSpacingLinesToPointse11111111111111111111111111111111lectionParagraphFormatLineSpacingLinesToPointsSelectionParagraphFormatLineSpacingLinesToPoctionParagraphFormatLineSpaci2222222222222222222222ngLinesToPoints2SelectionParagraphFormatLineSpacingLinesToPointsSelectionParagraphFormatLineSpacingLinesToPointselectionParagraph