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 :

at the same time!

( , ) 0

( , ) 0x

y

f x y

f x y

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

2 2( , ) 9f x y x y

Tangent plane x+y+z = 9

Challenge…•Where will the function

have critical points? Sketch this.

2( , ) cos ( )f x y xy

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

define:

( , ) 0 and ( , ) 0x yf a b f a b

2( , ) ( , ) ( , ) [ ( , )]xx yy xyD a b f a b f a b f a b

) 0 ( , ) 0 ( , ) is a local min

) 0 ( , ) 0 ( , ) is a local max

) 0 ( , ) is neither a local max or min

a if D and fxx a b f a b

b if D and fxx a b f a b

c if D f a b

Sample Questions…•Try 15.7: 2, 3, 7, 13,14,37, 47

Use Maple!15.7 #17