Question

Among all pairs of numbers with a sum of 238238, find the pair whose product is maximum. Write your answers as fractions reduced to lowest terms.

Answer 1

`x=119119,\;\;\;y=119119`

Explanation:

Let the
`x` be the first number,
`y` be the second number and
`p` be the product of the first and second number.

i.e
`x+y=238238`
`\Rightarrow x=238238-y`

`p=xy`

`\Rightarrow p=(238238-y)y\\\Rightarrow p=238238y-y^2`

`\Rightarrow (dp)/(dy) =238238-2y\\\Rightarrow 0=238238-2y\\\Rightarrow 2y=238238\\\Rightarrow y=119119`

Now,

`x=238238-119119\\\\ \Rightarrow x=119119`

Hence,
`x=119119,\;\;\;y=119119`