Differentiation Applications 2

**holmesb** · November 28th 2007, 02:00 AM

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.

**topsquark** · November 28th 2007, 02:39 AM

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
$x + y = 10 \implies y = 10 - x$

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

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

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

$4x - 20 = 0$

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

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

-Dan

**topsquark** · November 28th 2007, 02:41 AM

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
$x^2 + \frac{1}{x^2}$

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

-Dan

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