Maxima and Minima
An old friend with a new twist!
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 :
f x ( x, y )  0
f y ( x, y )  0
at the same time!
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
f ( x, y)  9  x2  y 2
Tangent plane x+y+z = 9
Challenge…
•Where will the function
f ( x, y)  cos ( xy)
2
have critical points? Sketch
this.
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
The 2nd Derivative Test…
If the 2nd partial derivatives are continuous
on a disk with center (a,b) and
f x (a, b)  0 and f y (a, b)  0
define:
D(a, b)  f xx (a, b) f yy (a, b)  [ f xy (a, b)]2
a ) if D  0 and fxx(a, b)  0  f (a, b)is a local min
b) if D  0 and fxx(a, b)  0  f (a, b)is a local max
c) if D  0 f (a, b) is neither a local max or min
Sample Questions…
•Try 15.7: 2, 3, 7,
13,14,37, 47
Use Maple!
15.7 #17