Question

X+y=50 which x and y are two positive numbers (a)for which numbers x is the product of the two numbers an increasing function of x (b)what is the maximum value of their product?

Answer 1

We can see that `y = 50 - x`. Let `p(x)` be the product.

`p(x) = xy = x(50 - x) = 50x - x^2`

Now we take the derivative.

`p'(x) = 50 - 2x`

Take the derivative at `0`.

`0 = 50 - 2x`

`2x= 50`

`x = 25`

Which means that the maximum product will happen when `x = y = 25`. This is a downward opening parabola which means anywhere left of the vertex is increasing, so the range is `0 < x < 25`.

Hopefully this helps!

Answer 2

See below.

Explanation:

Here the extremum point problem is equivalent to this one.

Given the function

`f(x,y) = x y - C = 0`

determine `C` such that `f(x,y)` and `x+y = 50` are tangent.

Choosing `x = C/y` and substituting into `x+y=50` we have

`C/y + y = 50` or

`y^2-50y +C = 0 rArr y = 1/2(50 pm sqrt(50^2-4C))`

but tangency implies one solution hence `50^2-4C = 0`

and finally

`C = 25^2, y = 25 rArr x = 25`

b) `25^2`