1. An old friend with a new twist!
Maxima and Minima
An old friend with a new twist!

2. Basic Conditions…. The slope of the tangent plane must be zero!
We can build a tangent plane out of the sum of two independent vectors so … f(x,y) is at a maximum (or min) if : at the same time!

3. Critical Points A point is critical if… fx and fy = 0
One of fx or fy (or both) fails to exist
Example: Find critical points on the surface
Tangent plane x+y+z = 9

4. Challenge…
Where will the function
have critical points? Sketch this.

5. Saddle Points…
Sometimes a critical point is not a max or a min. This is analogous to inflection points. Such points are called saddle points
pringle potato chip points

6. The 2nd Derivative Test…
If the 2nd partial derivatives are continuous on a disk with center (a,b) and define:

7. Sample Questions…
Try 15.7: 2, 3, 7, 13,14,37, 47
Use Maple! 15.7 #17