Optimization problem

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racuna

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Sep 28, 2005
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Find two positive numbers whose product is 100 and whose sum is a minimum.

Ok, I know that the formula for a product is P=x*y
So in this case, it would be 100=x*y
This is a type of related rates problem, so I have to find the sum,
x+y=___, but I don't know what the sum equals to.

Is there something that I am missing here?

I know after I have the number I have to make one equations in terms of one variable and then differentiate and find the critical numbers to find the minimum.

I just need the middle.
Oh, yes, the book gives the answers:10,10
 
Hint: If xy = 100, then what is an x-based expression for y?

Use this expression in your sum, and minimize.

Eliz.
 
Do you mean 100/x=y and x+(100/x) in the sum?

But what does the sum equal to?
 
You aren't given the sum to plug in; you have to find it.

It equals "f(x)", the function you now need to minimize. Find the minimizing value of x, and then back-solve for y and the sum.

Eliz.
 
Where are you stuck? You have a variable for the one number, an expression for the other number, and an expression for the sum. All that is left is the maximizing.

To "maximize" the function, differentiate and set the result equal to zero. Then solve for the value(s) of x. Check the function (the original one, not the derivative) at these values of x, and see which one gives you the biggest result.

Eliz.
 
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