CLASSROOM NOTE Some new mean value theorems of Flett type

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1 Interntionl Journl of Mthemticl Eduction in Science nd Technology CLASSROOM NOTE Some new men vlue theorems of Flett type Chenggun Tn nd Songxio Li Deprtment of Mthemtics Jiying University Meizhou Chin (Received August 013) In this pper we give some new men vlue theorems which re generliztions of Flett Myers nd Tong s theorems Keywords: Rolle s theorem; men vlue theorem; Flett s theorem 000 Mthemtics Subject Clssifiction: Primry 6A06; Secondry 6A4 1 Introduction Vrious men vlue theorems re importnt tools in mthemticl nlysis It is worth mentioning the pioneering contributions of Fermt Rolle Lgrnge Cuchy Tylor nd others The fmous men vlue theorem of Lgrnge ws stted s follows Theorem A (Lgrnge s Theorem): Let f(x) be rel continuous function on [ b] nd differentible on ( b) Then there exists point c ( b) such tht When f () f (b) then the men vlue theorem of Lgrnge reduces to Rolle s theorem which is nother importnt result in mthemticl nlysis The men vlue theorem of Lgrnge hs been generlized by mny uthors In 1958 vrition of the men vlue theorem of Lgrnge ws given by Flett [1] nd it ws lter extended in [ 4] nd generlized in [5 10] Theorem B (Flett s Theorem): Let f :[b] R be differentible on [ b] nd f () f (b) Then there exists c ( b) such tht In 1977 Myers [] proved in the sme condition tht there exists c ( b) such tht Emil: jyulsx@163com mth@jyueducn C 014 Tylor & Frncis

2 Clssroom Note In 1998 Shoo nd Riedel [10] gve generliztion of Flett s men vlue theorem tht no longer requires ny endpoint conditions Theorem C (Riedel nd Shoo s Theorem): Let f(x) be differentible function on [ b] Then there exists c ( b) such tht f (b) f () () () In 004 Tong [3] proved the following result Theorem D (Tong s Theorem): If f(x) is rel continuous function on [ b] nd differentible on ( b) then there exists c ( b) such tht Here M f () f (b) I 1 6(M I) () () f (t)dt In 01 Ckmk nd Tiryki [6] gve the following result which is slight modifiction of Riedel nd Shoo s Theorem ie if f (x) is differentible function on [ b] then there exists c ( b) such tht f (b) f () () () In this pper we give some men vlue theorems which re other generliztions of Flett nd Myers nd Tong s theorems This mteril might be used to enrich clculus or Introduction to Proofs course Min results nd proofs In this section we give nd prove our min results of this pper Theorem 1: If f(x) is rel continuous function on [ b] nd differentible on ( b) then there exists c ( b) such tht Here M f () f (b) N f () () ( b )

3 Proof Let Interntionl Journl of Mthemticl Eduction in Science nd Technology 3 g(x) f (x) f () x (x ) x (b) () Then g(x) is continuous on [ b b b] nd differentible on ( b) nd g (x) f (x)(x ) [f (x) f ()] (x ) () Next ( b g So g( b ) f ( b ) f () (M N) b b 4f ( ) 3f () f (b) g(b) which implies tht b f ( ) f () (M N) 4f ( b ) 3f () f (b) b ) g(b) By Rolle s theorem there is point c ( b) (b) such tht 0 () [] () () () () Similrly we cn get the following result Theorem : If (x) is rel continuous function on [ b] nd differentible on ( b) then there is vlue c ( b) such tht () () From Theorems 1 nd we get the following two corollries which show tht the ssertion of Flett s theorem holds in nother condition Corollry 1: If f(x) is rel continuous function on [ b] nd differentible on ( b) nd f ( b f ()f (b) ) then there exists c ( b) such tht

4 4 Clssroom Note Corollry : If f(x) is continuous function on [ b] nd differentible on ( b) nd f ( b f ()f (b) ) then there exists c ( b) such tht Theorem 3: Let f(x) nd g(x) be continuous functions on [ b] nd both differentible on ( b) If f (x)g (x)dx [f (b) f ()][g(b) g()] nd g (x) 0 for ny x ( b) then there is c ( b) such tht Proof Let h(t) t g(c) g() [f (x)g (x) f (x)g(x)]dx f ()g(t) g()f (t) Then h(t) is continuous on [ b] nd differentible on ( b) nd We hve h() 0 h(b) h (t) f (t)g (t) f (t)g(t) f ()g (t) g()f (t) [f (x)g (x) f (x)g(x)]dx f ()g(b) f (b)g() f (x)g (x)dx f (b)g(b) f ()g() f ()g(b) f (b)g() [f (b) f ()][g(b) g()] f (b)g(b) f ()g() f ()g(b) f (b)g() 0 So h() h(b) By Rolle s theorem there exists c ( b) such tht h (c) 0 ie Therefore we get f (c) g(c) f () g() 0 g(c) g() s desired In prticulr when g(x) x we get the following corollry which is the result of [3] Corollry 3: Let f(x) be continuous function on [ b] nd differentible on ( b) If 1 f (x)dx f (b) f ()

5 Interntionl Journl of Mthemticl Eduction in Science nd Technology 5 then there is c ( b) such tht Similr to the proof of Theorem 3 we cn get the following result Theorem 4: Let f(x) nd g(x) be continuous functions on [ b] nd both differentible on ( b) If f (x)g (x)dx [f (b) f ()][g(b) g()] nd g (x) 0 for ny x ( b) then there is c ( b) such tht g(b) g(c) Acknowledgements We wish to express our grtitude to the referees for their useful suggestions References [1] Flett T A men vlue theorem Mth Gz 1958;4:38 39 [] Myers R Some elementry results relted to the men vlue theorem Two Yer Coll Mth J 1977;8:51 53 [3] Tong J On Flett s men vlue theorem Int J Mth Educ Sci Technol 004;35: [4] Trhn D A new type of men vlue theorem Mth Mgzine 1966;39:64 68 [5] Abel U Ivn M Riedel T The men vlue theorem of Flett nd divided differences J Mth Anl Appl 004;95:1 9 [6] Ckmk D Tiryki A Men vlue theorems for holomorphic functions Electron J Diff Eqn 01;01(34):1 6 [7] Molnrov J On generlized Flett s men vlue theorem Int J Mth Mth Sci 01;01: [8] Pwlikowsk I An extension of theorem of Flett Demonstrtio Mth 1999;3:81 86 [9] Riedel T Sblik M A different version of Flett s men vlue theorem nd n ssocited functionl eqution Act Mth Sin 004;0: [10] Shoo P Riedel T Men vlue theorems nd functionl equtions River Edge (NJ): World Scientific; 1998