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
Noah G
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!

Cesareo R.
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#

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