Question

Find two positive numbers whose product is 25 and whose sum is a minimum. (if both values are the same number, enter it into both blanks.) (smaller number) (larger number)

Answer 1

Let the numbers be x and y:
xy=25
⇒y=25/x
the sum of the numbers is:
S=x+y
S=x+25/x
S=x+25x^-1
For minimum perimeter
S'(x)=0
thus
S'(x)=1-25x^-2=0
solving for x:
1-25x^-2=0
1=25/x^2
hence
x^2=25
thus
x=5
for manimum sum the numbers should be x=5 and y=5