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?

2 Answers
Mar 20, 2018

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!

Mar 20, 2018

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