How do you find two positive numbers whose sum is 300 and whose product is a maximum?

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Answer 1 · 2015-02-16T21:32:00

Let #x# and #y# be the two numbers whose sum is 300.
#y = 300 -x#

and let #f(x)# be their product
#f(x) = x xx (300 - x)#
#f(x) = 300x - x^2#

The maximum occurs when
#(d f(x))/dx = 0#

#(d (fx))/dx = 300 - 2x#
The minimum occurs when
#300 - 2x = 0#
#x = 150#

(and, since #x + y = 300#,
# y = 150#)