Question

Find two numbers whose product is 100 and whose sum is a maximum?

Answer 1

`xy = 100`

`y = (100)/(x)`


`x + y = A, A \in \mathbb{R}`

`x + (100)/(x) = A`

`(d)/(dx)(x + (100)/(x)) = 0`, since the differential of a constant is just 0.


`1 - (100)/(x^(2)) = 0`, to find the stationary points.

`1 = (100)/(x^(2))`

`x^(2) = 100`

`x = \pm 10`


`(d^(2))/(dx^(2))(x + (100)/(x)) = (d)/(dx)(1 - (100)/(x^(2)))`

`(d)/(dx)(1 - (100)/(x^(2))) = (200)/(x^(3))`

Thus, x = 10 will give you a maximum value, while x = -10 will give you a minimum value.

Thus, x and y = 10 will give you a maximum sum.

Answer 2

let them be X and Y..
now...
yx=100 ....
y=100/y ...
now...
x+y= s....where s is their sum...
but y=100/x
now....x+100/x=s..
by differentiating both sides with respect to x..
1-100/x^2=ds/dx
but ds/dx=0..
100=x^2 ..
x=10..
so y=10