定积分不等式

一．仅知被积函数连续的不等式1.设)(xf在1,0上连续且单调减少，证明：对于任意的1,0，都有100)()(dxxfdxxf。2.设)(xf在ba,上连续且单调减少，证明：babadxxfbadxxxf)()()(2。3.设)(xf在ba,上连续，证明babadxxfabdxxf)()()(22。4.设()fx在ba,上连续，且满足Mxfm)(0，证明：dxxfdxxfbaba)(1)(22)(4)(abmMMm-----------（Kantorovich）康托洛维奇不等式二．被积函数一阶可导的不等式（对)(xf用微分中值公式或对原函数)(xF用泰勒公式）5.设)(xf连续可微，且0)(10dxxf，Mxf)(，证明：任给(0,1)总有8)(0Mdxxf。6.设)(xf在ba,上连续可微，且1)(Mxf，0)(af，证明abMdxxfba21)(2)(。7.设)(xf在ba,上连续可微，且1)(Mxf，)(0)(bfaf，证明abMdxxfba21)(4)(。三．被积函数二阶可导的不等式（用泰勒公式）8.设)(xf在ba,上二次连续可微，且2)(Mxf，)(0)(bfaf，证明abMdxxfba32)(12)(。9.设)(xf在ba,上二次连续可微，且2)(Mxf，0)2(baf，证明abMdxxfba32)(24)(。10.设)(xf在ba,上二次连续可微，且0)(xf，证明2)()()()()2()(bfafabdxxfbafabba。四．具体函数的积分不等式11.证明2202021cos1sindxxxdxxx。12.证明2520sindxex。————————————————————————————————————一．仅知被积函数连续的不等式1.设)(xf在1,0上连续且单调减少，证明：对于任意的1,0，都有100)()(dxxfdxxf。证法1（换元法）令tx，10100)()()(dttfdttfdxxf（因f单减，)()(tftf）。证法2（单调性）设0)()(dxxfF，0)()()()()()()(220ffffdxxffF（单减f,,0）0))()(()()()(2020dxxffdxxffF或所以)(F单减，)1()(FF，即100)()(dxxfdxxf。证法3（利用定积分性质）0)()1()()1()()()1(,1,,,00)()1()()1()()()1()()()()()(1010001010ffdxxfdxxffffdxxfdxxfdxxfdxxfdxxfdxxfdxxf右或左单减右左证法4（函数最值）设1,0,)()()(100xdttfxdttfxFx1,0),()()()()(10fxfdttfxfxFx为函数的驻点，且x时)(,0)(),()(xFxFfxf单增，x时)(,0)(),()(xFxFfxf单减。所以)(F为函数的最大值，而最小值在端点取得。又0)1()0(FF，故0)(xF。(或者10)()()(dttfxfxF为单调递减函数，)(xF为上凸函数，最小值在端点取得。)证法5（微分中值定理）设1,0,)()(0xdttfxFx，则)()(,0)0(xfxFF)1,(),(1)()1(),0(),(0)0()(fFFfFF，因为f单减，所以)()(ff即1)()1()(FFF，也就是)1()(FF，得证。证法6（用定积分定义）ninnindxxfnifnnifndxxf11010)()(1lim)(lim)(2.设)(xf在ba,上连续且单调减少，证明：babadxxfbadxxxf)()()(2。证明作辅助函数xaxadttfxadtttfxF)()()(2)(，0)(aF。xadttfxfaxxxfxF)()()()(2)(0)()()()()(xaxadttfxfdttfxfax（因)(xf单调减少）所以)(xF单调减少，故0)()(aFbF，即不等式成立。3.设)(xf在ba,上连续，证明babadxxfabdxxf)()()(22。证明作辅助函数xaxadttfaxdttfxF)()()()(22，0)(aF)()()()()(2)(22xfaxdttfdttfxfxFxaxadtxfdttfdttfxfxaxaxa)()()()(2220)()(2xadttfxf，所以)(xF单调减少，故0)()(aFbF，即不等式成立。注此题也可用SchwartzCauchy不等式bababadxxgdxxfdxxgxf)()()()(222快速证明。同类的题还有：设)(xf在ba,上连续，且0)(xf，证明2)()(1)(abdxxfdxxfbaba。4.设()fx在ba,上连续，且满足Mxfm)(0，证明：dxxfdxxfbaba)(1)(22)(4)(abmMMm-----------（Kantorovich）康托洛维奇不等式证明：因为0)())()()((xfMxfmxf即MmxfmMxf)()(推得babaabMmdxxfmMdxxf))(()(1)(又babababadxxfdxxfmMdxxfmMdxxf)(1)(2)(1)(故)(2)()(1)(abMmdxxfdxxfmMbaba即dxxfdxxfbaba)(1)(22)(4)(abmMMm。注(特例)设()fx在[0,1]上连续，且满足1()2fx，证明：110019()()8fxdxdxfx。二．被积函数一阶可导的不等式（对)(xf用微分中值公式或对原函数)(xF用泰勒公式）5.设)(xf连续可微，且0)(10dxxf，Mxf)(，证明：任给(0,1)总有8)(0Mdxxf。证法一（对原函数用泰勒公式）记xdttfxF0)()(，1,0x，则0)1()0(FF，而,)(2)())(()()(2xtFxtxFxFtF1,0令1,0t，有,2)()()()0(021xFxxFxFF1,01（1）,)1(2)()1)(()()1(022xFxxFxFF1,02（2）（1）)1(x（2）x得)1()1)(()(21)(021xxxFxFxF则8)1(21)1()1(21)(MxMxxxxMMxxF。法二记xdttfxF0)()(，1,0x，则0)1()0(FF。若0)(xF，则结论自明。若)(xF不恒为零，则)(xF的最大值必在区间），（10内某点c取得，且该点为极值点，0)(cF。用泰勒公式可得,)(2)())(()()(2cxFcxcFcFxF故2)(2)()(cxMxFcF若2/10c，取0x可得82)(2McMcF，若12/1c，取1x可得8)1(2)(2McMcF。综上有8)(0Mdxxf。6.设)(xf在ba,上连续可微，且1)(Mxf，0)(af，证明abMdxxfba21)(2)(。证明法一dxafxfdxxfdxxfbababa)()()()(211)(2)()()(abMdxaxMdxaxfbaba。法二记xadttfxF)()(，0)()(aFaF，)()(xfxF2))((21)()()(abFaFaFbF所以)(bF212)(2)()(21abMabF。7.设)(xf在ba,上连续可微，且1)(Mxf，)(0)(bfaf，证明abMdxxfba21)(4)(。证记2bac法一仿3题可得2121)(8)(2)(abMacMdxxfca，2121)(8)(2)(abMcbMdxxfbc易知abMdxxfdxxfdxxfbccaba21)(4)()()(。法二令xadttfxF)()(，0)()(aFaF，)()(xfxF2))((21)()()(acFaFaFcF所以)(cF212)(8)()(21abMacF同理令bxdttfxG)()(，0)()(bGbG，)()(xfxG2))((21)()()(acGaGaGcG所以)(cG212)(8)()(21abMacG,从而abMcGcFdxxfba21)(4)()()(法三分部积分（用上条件)(0)(bfaf）babababadxxfcxdxxfcxabxfcxcxdxfdxxf)()()()()()()()()(故abMdxcxMdxxfcxdxxfbababa211)(4)()(三．被积函数二阶可导的不等式（用泰勒公式）8.设)(xf在ba,上二次连续可微，且2)(Mxf，)(0)(bfaf，证明abMdxxfba32)(12)(证法一分部积分可得（用上条件)(0)(bfaf）bababadxxfdxxfxbadxxfxbax)(2)()2()())((进而32)(12))((2)())((21)(abMdxxbaxMdxxfxbaxdxxfbababa。法二记2bac，分部积分可得bababadxxfcxcxdxfdxxf)()()()()(因为))(()()(cxfcfxfbababadxcxfdxcxfcfcxdxxf2))(())(()()()(故32222)(12)()()()(abMdxcxMdxfcxdxxfbababa注（特例）设)(xf在10，上二次连续可微，且2)(Mxf，)1(0)0(ff，证明Mdxxf12)(210。9.设)(xf在ba,上二次连续可微，且2)(Mxf，0)2(baf，证明abMdxxfba32)(24)(证明（涉及到端点外的第三点）记2bac，则0)(cf，在该点用泰勒公式有2))((21))(()(cxfcxcfxf，而bababadxcxfdxcxfcxcfdxxf22))((21))((21))(()(，abMdxcxMdxcxfdxxfbababa32222)(24)(2)()(21)(。10.设)(xf在ba,上二次连续可微，且0)(xf，证明2)()()()()2()(bf