中值定理在不等式证明中的应用 .pdf

.在泰勒中值定理证明不等式的应用中，给出了泰勒公式中展开点选取的几种情

况：区间的中点、已知区间的两端点、函数的极值点或最值点、已知区间的任意点.同

时对各种情况的运用范围和特点作了说明，以便更好的运用泰勒中值定理证明不等式.

并对柯西中值定理和积分中值定理在证明不等式过程中的应用问题作简单介绍•

关键词：拉格朗日中值定理；泰勒公式；柯西中值定理；积分中值定理；不等式

Abstract

Thispaperideawroteininequalityproofofusefrequentlyduringseveralofthe

meanvaluetheorem,whichintheLagrangemeanvaluetheoremprovinginequality

intheapplicationofthethreemethodstospeak:directformulamethod,variablevalue

method,themethodtoconstructauxiliaryfunction.intheapplicationofproofin

equalitiesoftheTaylormeanvaluetheorem,whichgaveTaylorformulaonthepoint

inseveralways:thepointoftheinterval,theintervaloftwoknownextreme,thefuncti

onextremevaluepointorthemostvaluepoint,theintervalofknownatanypoint.

Andtheapplicationrangeofofallkindsofsituationandcharacteristicsthatwere

explained,inordertobetteruseTaylorofthemeanvaluetheoremtotestifyin

equality.AndCauchymid-valuetheoremandintegralmeanvaluetheoreminthe

applicationprocesstoprovetheinequalitywerebrieflydiscussed

Keywords；

TheLagrangeMeanValueTheorerpTaylorsFormulaCauchyMean

:

ValueTheorem；Inequality；TheMeanValueTheoremforIntegrals

摘要()

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Abstract()

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1引言(1)

2拉格朗日中值定理在不等式证明中的应用(2)

2.1拉格朗日中值定理(2)

2.2利用拉格朗日中值定理证明不等式(2)

2.2.1直接公式法(2)

2.2.2变量取值法(4)

2.2.3辅助函数构造法(5)

3泰勒中值定理在不等式证明中的应用(7)

3.1泰勒中值定理(7)

3.2利用泰勒公式证明不等式(