by AuSmith » Fri Mar 23, 2007 3:19 pm
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.