In multivariable calculus, we were finding relative maxima and minima. It’s much like finding maxima and minima in 2D.
The general idea in 2D is that if you go a little bit to the left or a little bit to the right (changing x by a wee bit) at a maxima or minima, you aren’t really changing your height much (you aren’t changing y by much). Another way to look at it… if you zoom in enough to a maxima or minima, you’ll almost see a straight line! And you can make it as “straight” as you want it by zooming in more and more and more.
Does that make sense?
Now we do a similar argument for maxima and minima in 3D:
At the top of peaks or troughs, you’ll notice if you walk a wee little bit in the x direction, the height (z) isn’t changing by much. Similarly if you move a wee little bit in the y direction, the height isn’t changing as much. (Or, analogously, if you zoom in a lot lot lot lot, you’ll be looking at something almost perfectly flat, a horizontal plane…)
In other words, instead of saying maxima and minima only occur when , we now can say that maxima and minima only occur when and . That’s the mathematical way to talk about moving a bit in a x-direction or y-direction.
So my kids know to find possible relative maxima or minima, you have to find the points which make and .
In class I then posed a few good questions:
(a) If you know a maximum occurs at the point , how can you show that the directional derivative in the direction is also 0?
(b) If you know that at the point , the directional derivative for is 0, and the directional derivative for is also 0. Prove that the point is a maximum or minimum or saddle point.
These were things that I thought up on the fly… it’s interesting. We get so used to procedures, that we sometimes forget what they mean. The point I was trying to make is that if any two (different) directional derivatives were 0 at a point, then that point could be a maxima or minima. If you pose that as a claim, and students are used to thinking algebraically, they have to go through the motions to see this is true. (It basically involves creating and solving a system of two linear equations…). 
But there’s a much easier way to get students to buy that claim. If you think graphically, this makes sense… if you are at a maxima or minima, and you zoom in enough, the surface will look like a flat plane. So of course if you walk a short distance in any direction, you shouldn’t be moving (much) in the direction.
I don’t know… this isn’t deep or anything. But it was something that I didn’t plan in class that I thought was interesting…
 This is how it would go… Assume you know and (and the two vectors aren’t scalar multiples of each other). Then you can rewrite and . Well then you simply have a system of equations that you can solve for and — and it is easy enough to show that the solution is and .