Where do extrema occur?
First, you need to know where to look for the max or min of a function.
There are three cases.
1. At a boundary point.
Remember doing this in 2-d? Indeed it was many moons ago, but its basically the same idea. In the function f= x^2, from x= -2 to 2, the endpoints are -2 and 2. Both ends of the line or curve. In 3-d, it’s the boundary edge - just like a balls surface or a country’s border. In context with the other two, it make a bunch of sense. The max point should be at the begining, somewhere in the middle, or at the end right? Yes, and in 3-D it’s the edge/surface of the boundary.
2. Stationary point
A stationary point is some point at which the functions tangent is horizontal, ie: the gradient operator (the upside down triangle) = 0. To see it visually, imagine that the function f models an anthill. Imagine taking a glass plate, and while holding it completly level, lower it down until it just barely touches the antill. The point at which the first ant is squished is the highest point (max). You can remember this because the ant becomes stationary when you smash the life out of him (with calculus).
The same thing would be true if your friend held a basketball and you started beneath the ball raising it up until it first touched the bottom (min).
3. Singular points
This is some interior point (ie: not on the boundary) where the function is not differentiable. See my section (to be completed soon!) about differentiability and contunity.
So - in recap the trinity of max/min locations.
Boundaries
Stationary
Singular
Sweet.
NOW THEN, how do you find out if a point is a max, or a min in 3-D space?
I’m glad you asked.
Recall the second derivateve test for functions of one variable? No?
No problem, I’m your pusher man.
I call it the D test, but your welcome to call it whatever you’d like.
It invovles computing the Fxx (partial x derivative twice) , the Fyy( the partial y twice) and the Fxy partial derviatives and multiplin’ em together and what not. So just compute the partials, and save ‘em for analysis. You’ll get a number and then you just see what the number says about the function. See sarcastic charts below.
OFFICIAL DECLARATION
- see page 672 -
