From: rusin@vesuvius.math.niu.edu (Dave Rusin)
Newsgroups: sci.math
Subject: Re: Derivate
Date: 19 Oct 1996 04:37:34 GMT
In article <543v2t$3lt@news.dgsys.com>, Randy Poe wrote:
>
>>A 2 meter ironwire can be bent like a circle or like a square or be cut in
>to
>>peaces and bent like a square and a circle. How much wire will be used in
>>the circle if the total area of the square and circle is to be minimal.
> ^^^^^^^
>
>A function is minimized when its derivative is zero. So if we can
>express the total area in terms of our variable, all we need to do
>is take the derivative.
Yes, and that's how a function is maximized too. But this classic
question yields only one zero to the first derivative, which cannot
be the location of both the max and the min!
Students need always remember the whole picture:
If f achieves a (local) min (or max) at a point a, then either
1. f'(a) = 0
2. f'(a) does not exist or
3. a is an endpoint (i.e. not an interior point of the domain
over which f is being optimized).
Moreover,
4. If (1) (2) or (3) holds, f need not have a min or max at a
(not even a local min/max)
5. But f will attain a min and a max somewhere on its domain if
that domain is closed and bounded and f is continuous.
The wire problem is used to see if students are remembering anything
besides (1). I like similarly to give problems which read "maximize
profit subject to these constraints..." and the students proceed to
_minimize_ the profit! It's easy to realize the importance of a
theorem when you see you might get fired as a result of doing it wrong.
dave