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Derivata a treia nenegativa (interpretare geometrica)

 
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Mateescu Constantin
Newton


Joined: 21 Apr 2009
Posts: 398
Location: Londra/Pitesti

PostPosted: Tue Mar 13, 2012 7:49 pm    Post subject: Derivata a treia nenegativa (interpretare geometrica) Reply with quote

Aratati ca daca o functie are derivata a treia nenegativa pe un interval, atunci panta dreptei determinata de doua puncte de pe graficul corespunzator acelui interval nu depaseste media aritmetica a pantelor tangentelor la graficul functiei construite in cele doua puncte.

Cu alte cuvinte, daca f este o functie de trei ori derivabila pe un interval I cu f^{\prime\prime\prime}\, \ge\, 0 pe intervalul I,

atunci are loc urmatoarea inegalitate: \fbox{\ \frac {f(x)-f(y)}{x-y}\, \le\, \frac {f^{\prime}(x)+f^{\prime}(y)}2\ }\ \ ,\ \left(\forall\right)\ x\, ,\, y\, \in\, I\ ,\ x\, \ne\, y .
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DrAGos Calinescu
Thales


Joined: 07 Dec 2008
Posts: 142
Location: Pitesti

PostPosted: Wed Mar 14, 2012 4:53 pm    Post subject: Reply with quote

\frac{f(x)-f(y)}{y-x} se poate chiar incadra, fiind si cel putin egala cu  f^{\prime}( \frac{x+y}{2})
Reiese din inegalitatea Hermite-Hadamard aplicata functiei convexe f^{\prime}
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