Question

Question #58f16

Answer 1

the numbers are both 5

Explanation:

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.