6.3
Optimazation
Consider the continuous function of two variables, defined on a closed bounded region The values ( ) and ( ) where ( ) ( ) ∈ such that
( )≤ ( ) ≤ ( )
for all ( ) ∈ are called the minimum and maximum of in the region
Theorem 70 Let be a continuous function of two variables and defined on a closed bounded region in the −plane.
1. There is at least one point in where takes on a minimum value. 2. There is at least one point in where takes on a maximum value.
A minimum is also called an absolute minimum and a maximum is also called an absolute maximum.
Definition 32 (Relative Extrema) Let be a function defined on a region containing (0 0)
1. The function has a relative minimum at (0 0) if
( )≥ (0 0)
for all ( ) in an open disc containing (0 0)
2. The function has a relative maximum at (0 0) if
( )≤ (0 0)
for all ( ) in an open disc containing (0 0)
Definition 33 (Critical Point) Let be defined on an open region con-taining (0 0) The point (0 0) is a critical point of if one of the
fol-lowing is true.
1. (0 0) = 0 and (0 0) = 0
2. (0 0) or (0 0) does not exists.
Definition 34 (Saddle point) A saddle point is critical point that is neither a high point nor a low point on the graph of a function of two independent variables.
Theorem 71 If has a relative extremum at (0 0) on an open region
then (0 0) is s critical point of
Example 158 The lowest point on graph of
( ) = 70− −2−2+4−25
is the critical point. Find the value of and the value of where ( ) minimum. Then find the minimum value of ( ) (≈ 154)
Example 159 Find the critical point of ( ) = 23+ 1
3
3
− 2− 24 − 3 Example 160 Find the critical point of
( ) = 83+ 3− 6 + 5
