The product of two positive real numbers x and y is 20. Find the minimum possible value of their sum.If you could explain it simply, I'd like to learn how to do it for myself, rather than just...

The product of two positive real numbers x and y is 20. Find the minimum possible value of their sum.

If you could explain it simply, I'd like to learn how to do it for myself, rather than just obtaining the answer. Thanks!

Asked on by shizuo

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hala718 | High School Teacher | (Level 1) Educator Emeritus

Posted on

Let the numbers be x and y

Then the product of x and y is:

x*y = 20

Now we need to determine th minimum of their sum

Let f(x) = x + y

But x*y = 20 ==> y = 20/x

==> f(x) = x + (20/x)

Now to find the extreme value we will differentiate the function:

==> f'(x) = (x)' + (20/x)'

= 1 - 20/x^2

Now we will determine the derivative's zero's:

==> 1- 20/x^2

==> 20/x^2 = 1

==> x^2 = 20

==> x  = sqrt20 = 2sqrt5

Now we will find the second derivative:

f''(x) = 0 + 20/x^4

Since the second derivative is positive, then the function has a minimum point.

Now we determined the value for x

but y = 20/x = 20/ 2sqrt5

= 10/sqrt5 = 10sqrt5/5 = 2sqrt5

Then y= 2sqrt5

Then the minimum sum for the numbers is:

x + y = 2sqrt5 + 2sqrt5 = 4sqrt5

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