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Question Number 140734 by liberty last updated on 12/May/21

Let m,n be given positive integers.  If x & y are positive numbers such that  x+y= S , S is a constant, find  the value of x and y that maximize  Q=x^m y^n  .

Letm,nbegivenpositiveintegers. Ifx&yarepositivenumberssuchthat x+y=S,Sisaconstant,find thevalueofxandythatmaximize Q=xmyn.

Commented bymr W last updated on 12/May/21

if m and n both are even, there is   minimum.    if only one of m, n is even, there is no  maximum and no minimum.    if both m and n are odd, there is   maximum.

ifmandnbothareeven,thereis minimum. ifonlyoneofm,niseven,thereisno maximumandnominimum. ifbothmandnareodd,thereis maximum.

Answered by EDWIN88 last updated on 12/May/21

setting (dQ/dx) = mx^(m−1) (S−x)^n −n(S−x)^(n−1) x^m  = 0  we obtain x = ((mS)/(m+n)) . The first−derivative test  show that this yields a relative maximum   and therefore by the uniqueness of critical  number, an absolute maximum when x=((mS)/(m+n)) , y=((nS)/(m+n))

settingdQdx=mxm1(Sx)nn(Sx)n1xm=0 weobtainx=mSm+n.Thefirstderivativetest showthatthisyieldsarelativemaximum andthereforebytheuniquenessofcritical number,anabsolutemaximumwhenx=mSm+n,y=nSm+n

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