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Answer 1 · 2018-01-22T02:59:11

#5#

let #x=#one of the positive integers
that means #10-x# is the other positive integer since they add to 10.

the product of those numbers is #x(10-x) = -x^2+10x#
you want to maximize the value #-x^2+10x#

this graph is a parabola opening down, as you can see from the -1 coefficient on the #x^2# term.

the maximum value of #-x^2+10x# occurs at the vertex, or where the derivative = 0.

#d/dx(-x^2+10x)=-2x+10#
#-2x+10=0# when #x=5#

this means the vertex of the graph is #(5,25)#, meaning the two positive integers are 5 and 10-5, or simply 5 and 5.

the answer is #5#.