A Milk Man Doth Procrastinate

[AuSmith]

(1) Leia Z is told to bring home a Canadian muskrat for dinner. If Canada is the region [unparseable or potentially dangerous latex formula], Leia starts at [unparseable or potentially dangerous latex formula], and home is [unparseable or potentially dangerous latex formula], what's the shortest distance Leia can travel if Canada is saturated with easy-to-kill muskrats?

(2) A point mass milk man named Bo Vinum has a bad habit of procrastinating. He just woke from a nap on the point [unparseable or potentially dangerous latex formula], and he must milk the poor swollen cow on [unparseable or potentially dangerous latex formula] quickly if he is to make his date at the point at infinity with the handsome seamstress next door. To make matters worse, before he can get to milking, he has to wash the bucket in the nearby stream which sits on the line [unparseable or potentially dangerous latex formula]. What is the shortest route Bo can take to reach his cloven friend?

(3a) What're the maxima and minima (if they exist) of the function [unparseable or potentially dangerous latex formula]?

(3b) What are the extrema of [unparseable or potentially dangerous latex formula] if we only consider points on the circle [unparseable or potentially dangerous latex formula]?

[bradp]

nevermind

[suchenzang]

I don't think that will give you the shortest distance. If you look at it graphically and you move the point where your line intersects the x-axis closer to the origin, you will find that the total distance is gradually decreasing and then increasing again.

Looking at the picture, I would guess that your minima would occur at x=2. You can confirm this by calling the distance between your point C and (0,0) x. Then add the two squares of the distances AC and BC and minimize your equation. I'm assuming that when you minimize the sum of two squares you're minimizing the numbers themselves...

Here's the math if you want to see it:
[unparseable or potentially dangerous latex formula]
[unparseable or potentially dangerous latex formula]
[unparseable or potentially dangerous latex formula]
Minimize:
[unparseable or potentially dangerous latex formula]
[unparseable or potentially dangerous latex formula]

edit: oops. completely used fallacious logic there. minimizing the sum of two squares of positive numbers doesn't minimize the sum of the two numbers themselves. dumb, dumb, dumb...

[Kurt]

[suchenzang]

oops, i misread the one of the points as 4,-14 instead of 7,-14

[AuSmith]

[Kurt]

3.5 came from me being lazy and therefore attempting to avoid doing the derivative, and believing in the power of symmetry. :P Of course, I forgot that -10 and -14 aren't the same number (:P) and so it's not symmetric anyway.

So from (0,-10) to (7,-14)...with the x-axis somewhere in between, I guess I need to solve:

[unparseable or potentially dangerous latex formula]
[unparseable or potentially dangerous latex formula]

After a bit of algebra...[unparseable or potentially dangerous latex formula]

So [unparseable or potentially dangerous latex formula]
Hmm, an exact answer.

[AuSmith]

[AuSmith]

Sail away sail away

[AuSmith]

[AuSmith]

Ok, we found the minima on #3 fairly easily and quickly. Now, find the maxima (part b). It might use something you haven't seen, so don't beat yourself to death with things you already know.

[makashi]

By the Cauchy-Schwartz inequality,
[unparseable or potentially dangerous latex formula]
[unparseable or potentially dangerous latex formula]
The maximum of [unparseable or potentially dangerous latex formula] occurs at [unparseable or potentially dangerous latex formula].

[AuSmith]

That's right.


If you know about partial derivatives, here's another solution to part (b). First I will define [unparseable or potentially dangerous latex formula] (read "del f"). It's the vector

[unparseable or potentially dangerous latex formula]

To find the critical points of [unparseable or potentially dangerous latex formula] under the constraint [unparseable or potentially dangerous latex formula], set [unparseable or potentially dangerous latex formula] in the same direction as [unparseable or potentially dangerous latex formula].

Here, we are trying to maximize [unparseable or potentially dangerous latex formula] under the condition that [unparseable or potentially dangerous latex formula].


[unparseable or potentially dangerous latex formula]

[unparseable or potentially dangerous latex formula]

Now, set them in the same direction. As in, del f is a scalar multiple of del g.


[unparseable or potentially dangerous latex formula]

Now, we have 3 equations and 2 unknowns.

[unparseable or potentially dangerous latex formula]
[unparseable or potentially dangerous latex formula]
[unparseable or potentially dangerous latex formula]

The solutions of this system are your critical points. Substitute 2x-4y in the third equation.

[unparseable or potentially dangerous latex formula]

And we eliminated c.

[unparseable or potentially dangerous latex formula]

And, we now have the two points that we already knew: [unparseable or potentially dangerous latex formula] and [unparseable or potentially dangerous latex formula]. The first is the maximum and the second is the minimum.

[Kurt]

Oh, hahaha I had forgotten it asked for extrema...that's why I only gave the minimum.

[AuSmith]

Find the maximum and minimum volumes of a rectangular box whose surface area is [unparseable or potentially dangerous latex formula] and whose total edge length is [unparseable or potentially dangerous latex formula]. You may use Lagrange multipliers again.