1. Extrema of Functions of Two Variables using MATLAB
Isaac Bryan, Taryn Roten, Kimberly Kiser, and Kaylee Kemper

2. Applications
Rigid Body Dynamic
Engineering
Economics

3. More Applications
World Trade
Operational Research

4. Optimization
Optimization is most commonly defined as finding the largest or smallest possible values of a function. One way to find a maximum/minimum value, is to take a derivative of the function, find all critical points, evaluate the value(s) at those points, and discover which is a maximum/minimum value by using the Fundamental Theorem of Calculus.

5. Using more than 2 variables
In using 2 or more variables, the concept of optimization does not change. The derivative is taken with respect to each variable brought into the function. The critical points are then found within the function from the derivatives.
For two variables, a critical point of a function f defined on an open set D is a point (x,y) in D where either
At least one of fx(x,y) or fy(x,y) doesn’t exist at (x,y)
fx(x,y) = fy(x,y) = 0

6. Using more than 2 variables (cont.)
The discriminant of f is the expression
D(x,y) = fxx(x,y) fyy(x,y) - fxy²(x,y)
D(x,y) f xx(x,y) Type
Rel. Max.
Rel. Min.
- NA Saddle Point
0 NA Inconclusive

7. Examples Relative Max. Relative Min. Saddle Point z=1-x²-y² z=x²+y²
z=y²-x²

8. How our Program Works
The program generates a specified number of random values to start the program
The random values are placed in a loop to approach the extrema by “walking up the gradient”
The path the program took to reach the “absolute” extremum is plotted along with the function

9. Testing Our Program One Extremum Relative Max. & Min. Multiple Maxima
z=1-x²-y²
Relative Max. & Min.
z=9x/(x2+y2+1)
Multiple Maxima
z=cos(x)cos(y)
Saddle Points
z=y²-x²

10. Example of Testing Process
⇒ critical points are (-1,0) & (1,0)

11. Example of Testing Process (cont)
⇒ D(1,0)=20.25 ; fxx(1,0)=-4.5 And D(-1,0)=20.25 ; fxx(-1,0)=4.5 Therefore (1,0,4.5) is a Relative Maximum And (-1,0,-4.5) is a Relative Minimum

12. Issues with the Technique
Existence of a maximum
Multiple maxima
Global Maximum vs. Relative Maximum

13. Goals for Next Semester
Multiple Variables
Fewer Test Points
Faster Process

14. Cool Functions