Question

show that the product of three positive number of constant sum is maximum when they are equal

Answer 1

Answer:

Step-by-step explanation:

Suppose the (say, positive) sum of the numbers is S, and we suppose that each of the summands is required to be positive (otherwise, by taking two of the summands to be large and negative and the third positive, we can make the product as large as we like).

If

x,y,z

are three numbers whose sum is S, then z=S−x−y, so the product of the three numbers is

P=xyz=xy(S−x−y),

which we hence regard as a function of (x,y) in the first quadrant {x,y>0}.

Where this function has a local extremum, we have

0=∂P∂x=Sy−2xy−y2,

and by symmetry

0=∂P∂y=Sx−2xy−x2.