Question

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

Answer 1

Suppose we have a constant number `c` composed of the two numbers `x`, and `y`, That is:

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

Then the product, `P`, of the two numbers, is:

`P = xy`
`\ \ \ = x (c-x) \ \ \ \ \` (using [A])
`\ \ \ = cx-x^2`

Then we differentiate wrt `x` to get the first and second derivatives:

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

And we look for a critical point by looking for values of `x` so that the first derivative vanishes, thus we require:

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

`:. x=c/2`

And, using [A] with `x=c/2`, we get:

`c/2+y=c => y=c/2`

Noting that at this critical point the second derivative `lt 0` then we can conclude that we get a maximum when:

`x = y = c/2`

ie when they are equal, QED