1. ## Minimum

Find two numbers whose difference is 100 and whose product is a minimum.

The problem was given without a function and so I'm not really sure what to do.

2. Originally Posted by bobsanchez
Find two numbers whose difference is 100 and whose product is a minimum.

The problem was given without a function and so I'm not really sure what to do.
$a-b=100$

$ab=minimum$

$a=100+b\ \Rightarrow\ b=a-100$

$ab=a(a-100)$ in terms of "a".

Differentiate with respect to "a" and equate to zero to find "a" giving minimum product.
Then obtain "b".

3. So...

a= 50

b= -50

4. Originally Posted by bobsanchez
So...

a= 50

b= -50
Yes,
that's it.

5. Another problem similar to that one...

Find two positive numbers whose product is 81 and whose sum is a minimum.

I've started it with the method you suggested but I don't know where to go from a = 81/b. Is this where I differentiate?

6. Originally Posted by bobsanchez
Another problem similar to that one...

Find two positive numbers whose product is 81 and whose sum is a minimum.

I've started it with the method you suggested but I don't know where to go from a = 81/b. Is this where I differentiate?
Yes, you are forming an equation in one variable.
Differentiate the sum and equate to zero and solve.

7. I get 0.

8. Since the problem asked you to find two numbers, I have no idea what you could mean by "I get 0".

What function are you differentiating? The problem asked for two numbers "whose sum is a minimum". The sum of two numbers, x and y, is f(x,y)= x+ y. That is what you want to differentiate. You are also told that their product is 81: xy= 81 so y= 81/x so that $f(x)= x+ 81/x= x+ 81x^{-1}$.

9. Ah, I see what I did. I tried to differentiate y = 81/x.

Alright, so the answer is x = 9 and y = 9?

10. Originally Posted by bobsanchez
Ah, I see what I did. I tried to differentiate y = 81/x.

Alright, so the answer is x = 9 and y = 9?
Ok,
you need to understand the point of having 2 clues.
One clue allows you to write one variable "in terms of the other".

Then you can write an equation in one variable only.

y=81/x is only one component of the sum equation.
You want the sum to be a minimum, but you must work with the entire sum formula.

xy=81
x+y=sum.

y=81/x

Therefore x+81/x=sum.

$x+81x^{-1}=\ sum.$

sum is a minimum, so the derivative, wrt x, of the sum is zero.

See?

11. Yeah, that makes sense to me. I just have trouble seeing the relation between the two and putting them in a form I can work with to solve the problem.