# Prove that product of two parts of a constant number will be maximum when both parts are equal?

##### 1 Answer

Jun 13, 2018

Suppose we have a constant number

# x+y=c# ..... [A]

Then the product,

# P = xy #

# \ \ \ = x (c-x) \ \ \ \ \ # (using [A])

# \ \ \ = cx-x^2 #

Then we differentiate wrt

# (dP)/dx = c-2x # and# (d^2P)/(dx^2) = -2 #

And we look for a critical point by looking for values of

# (dP)/dx = 0 => c-2x = 0#

# :. x=c/2 #

And, using [A] with

# c/2+y=c => y=c/2#

Noting that at this critical point the second derivative **maximum** when:

# x = y = c/2#

ie when they are equal, QED