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Answer 1 · 2017-07-11T13:00:14

the numbers are both 5

let #x# and #y# be the positive numbers

#x+y=10#

Let #x^2+y^2=z#
#z# is a function of #x# and #y# which can be plotted as a graph.

For the sum, #z#, to be minimum:
we need to find its minimum point.

A minimum point on a graph has a gradient of 0.
#:.dz/dy=dz/dx=0#

from #x+y=10#,
we have:
#x=10-y#

Substitute the equation into #x^2+y^2=z#
#z=(10-y)^2+y^2#
#z=100-20y+y^2+y^2#
#z=2y^2-20y+100#

We use #dz/dy=0# since we have #z# in terms of #y#
#dz/dy=4y-20=0#
#4y=20#
#y=5#

We have #y#, to find #x#, substitute #y=5# into #x=10-y#
#x=10-5#
#x=5#

Note: you can have #y# in terms of #x# and then solve the problem with the same approach as I use, you will get the same answer.