Arithmetic Mean and Geometric Mean

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1 Acta Mathematca Ntresa Vol, No, p ISSN Arthmetc Mea ad Geometrc Mea Mare Varga a * Peter Mchalča b a Departmet of Mathematcs, Faculty of Natural Sceces, Costate the Phlosopher Uversty Ntra, Tr A Hlu, SK Ntra b Naroda baa Slovesa, Imrcha Karvasa, 83 5 Bratslava Receved October 06; receved revsed form October 06; accepted 4 October 06 Abstract I mathematcs we defe several types of meas Probably the most ow are the arthmetc ad geometrc meas If we are gve oegatve umbers,, ; the the umber A we call the arthmetc mea ad the umber G we call the geometrc mea of the umbers gve I the frst part of the paper wth the use of fuctos ther of more varables we wll show that for each atural umber apples I the secod part we wll try to show the same, however, wthout usg the A G dfferetal calculus Keywords: arthmetc mea, geometrc mea, proof, dfferetal calculus Classfcato: E55 Itroducto Whe gve oegatve umbers,, The the umber A we call the arthmetc mea of the umbers,, Number G we call the geometrc mea of umbers,, We wll try to fd the relato betwee umbers A a G Problem A Dvde the postve umber to two addeds so ther product would be the hghest the possble Soluto (See ) Lets dvde umber to two parts, whch wll be labeled as a y (0 ; 0 y ) Product of ths two addeds should be as hgh as possble, that s we search the global mamum of the fucto s(, y) = y; whle + y = apples We ca see, etreme of the two varables fucto should be foud s = s(, y) However, t wll be easer to calculate the global mamum of the oe varable fucto s = s() = ( ); where 0; Whe solvg the equato s() = 0 we wll get the statoary pot u0 Wth the help of the secod * Correspodg author; emal: mvarga@ufs DOI: 07846/AMN

2 44 Acta Mathematca Ntresa, Vol, No, p dervato s() ad comparso of the value s wth values s(0), s() we wll fd out, that the fucto s() acqures ts hghest value pot u0 It meas, that the fucto s(, y) acqures global mamum the pot X 0 ; So f 0, 0 y, + y =, the y y Thece mples, that y y, ths s G A Problem A Dvde the postve umber to three addeds so ther product would be the hghest the possble Soluto (See ) Lets dvde umber to three parts, whch wll be labeled as, y a z (0 ; 0 y ; 0 z ) Product of ths three addeds should be as hgh as possble, that s we search the global mamum of the fucto s(, y, z) = yz; whle + y + z = apples Aga, we wll mae the problem easer we wll search global mamum of the two varable fucto s(, y) = y( y) the doma restrcted by the les = 0, y = 0, + y = Whe solvg the scheme of equato s s y, y 0, y 0 we wll get statoary pots U0 ; ; U 0; ; U ; 0 ; U3 0; It wll be easly possble to show that the fucto s(, y) acqures ts hghest value at the pot U 0 ; 3 3 It meas that the fucto s(, y, z) acqures global mamum at the pot X 0 ; ; Therefore f 0, 0 y, 0 z, the follows that 3 3 y z yz apples Thece t 3 y z yz, that s G3 A3 3

3 Mare Varga Peter Mchalča: Arthmetc mea ad geometrc mea 45 Problem A3 Fd the hghest value of the etracto of the postve umbers,, product whe codtoed that the sum of these umbers s equal to umber Soluto (See ),, I the prevous sum we have foud the mamum of the fucto s, whle = apples We say that by the equalty = the bod s gve, that s we calculate the fed etreme of the fucto s(,, ) We wll set the Lagrage fucto,,, L Statoary pots wll be foud whe solvg the system of equatos L,,, 0 L,,, 0 we obta L,,, 0 L,,, 0 L 3 L 0 L L L 0 L 3 L L 0 L L 0 Sce, the equalty have to apply Thece out of L the equato 0 t follows that It could be verfed, that the

4 46 Acta Mathematca Ntresa, Vol, No, p s,, acqures at the pot X0 ; ; ; fucto meas fed global mamum That apples, or G A Problem B Prove that for, R : Soluto (See 3) Probably 0 The 0 < < ( ) + > + due to 0 < <, what had to be show Problem B Prove that for R +, =,,,, ad Soluto = = apples (See 4) The sum wll be proved by mathematcal ducto: a) = : trvally apples; b) = : apples accordg to the sum B, cosder: If the ether = = ad the sum s trval aga ad, or / / ad wthout ay loss o geeralty t s possble to suppose that The eg (0 ) The accordg to B t s obvous that, however whe, the Whe combg the trval ad the geeral parts we get the affrmato the form for = : c) Lets follow the ducto geerally Whe for, trvally (wth the character of equalty) Let for R +, =,,,, holds +, the affrmato apples Assume ow that for y j R +, j =,,, +, holds j= y j = Wthout ay loss of geeralty of the sum we wll relabel the umbers, so y ad y The ( y )( y ) 0, or + y y y + y, ad we obta + y y + y y + y + y + y y +

5 Mare Varga Peter Mchalča: Arthmetc mea ad geometrc mea 47 + Mar = y y, = y 3,, = y + The = = j= y j = O base of ducto hypothess The we have = + = = y y + j=3 y j ad so + y y + j=3 y j + But y + y + y y, so y + y + j=3 y j +, fally j= y j + Problem B3 Prove that for: R G A (thus the pre =,, platí : geometrc mea G of postve real umbers s smaller or at a most equal to the arthmetc mea A of these umbers) Soluto (See 3) If the equalty that have to be proved wll be smply tured to the form ad multpled by the term y Probably, we wll get Lets label y, where y 0 for, Fally, the reader wll cosder that y, so the umbers y meet the requests from the sum B thas to what the proved equalty G A apples Cocluso Usg two dfferet strateges (based o applyg of dfferetal calculus of fucto of more varables ad usg mathematcal ducto too) we have show a well ow equalty betwee geometrc mea ad arthmetc mea We are sure there could be llustrated further eteso ad geeralzato of ths relato Namely more comple equaltes could be demostrated usg college owledge, such as H G A K, where harmoc mea ad K s quadratc mea H s

6 48 Acta Mathematca Ntresa, Vol, No, p Refereces MAREK VARGA: Zbera úloh z matematcej aalýzy II; FPV UKF Ntra, 005, ISBN PETER MICHALIČKA: e-lear urz 3 ZBYNĚK KUBÁČEK, JÁN VALÁŠEK: Cvčea z matematcej aalýzy I, UK Bratslava, 994, ISBN [4] ALOIS KUFNER: Nerovost a odhady, ŠMM, Mladá Frota, Praha, s

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