Story Problems Using Derivatives?

Question

Find Two Positive Numbers whos product is 100 and whose sum is a minimum.

I performed a chart to get the numbers but again. The equation eludes me. All I could get was xy=100 and for the equation I have x+y=0 , other than that I'm lost!

Biznachos..question tho...where does the 1 go? 1-(100/x²) ?

Answer 1

x*y = 100

x+y does NOT equal 0.

if y=(100/x), then x+y = x + (100/x)

take the derivative, and set it equal to zero.

1 - (100/x^2) = 0

so x^2 = 100 and x = 10

since x*y=100, y = 10 also.

Remember, you can't just set equations equal to zero. You have to come up with an expression, and then take a derivative first.

Answer 2

It's a lot easier to minimize things when there's only one variable, so let's go immediately to that kind of formulation:

We want to minimize x + 100/x.

Well, do it in the usual way. Take the derivative of that function, set it equal to 0, and see where you are.

1 - 100/x^2 = 0

That solves to x = 10.

How do you know that's a minimum? Well, look at the second derivative -- it turns out to exist and to be positive whenever x is anywhere near 10.

So yes, the minimum is when both numbers are equal to 10.