1. ## Differentiation Applications 2

I have two more quick questions:

The sum of two numbers is 10. What are the two number if the sum of their squares is a minimum?

What numbers, when squared and added to their own squared reciprocals, lead to a minimum value?

Help is greatly appreciated.

2. Originally Posted by holmesb
I have two more quick questions:

The sum of two numbers is 10. What are the two number if the sum of their squares is a minimum?
Let the numbers be x and y. Then
$\displaystyle x + y = 10 \implies y = 10 - x$

We wish to minimize $\displaystyle f(x, y) = x^2 + y^2$

Well,
$\displaystyle x ^2 + y^2 = x^2 + (10 - x)^2 = 2x^2 - 20x + 100$

Critical points of this are at:
$\displaystyle \frac{d}{dx}(2x^2 - 20x + 100) = 0$

$\displaystyle 4x - 20 = 0$

$\displaystyle x = 5$
(I leave it to you to prove that this is a minimum.)

Thus $\displaystyle y = 10 - 5 = 5$ as well.

-Dan

3. Originally Posted by holmesb

What numbers, when squared and added to their own squared reciprocals, lead to a minimum value?
Call the number x. The we want to minimize
$\displaystyle x^2 + \frac{1}{x^2}$

So take the derivative, set it equal to 0 and solve for x.

-Dan