1. Applied Calculus, 3/E by Deborah Hughes-Hallet Copyright 2006 by John Wiley & Sons. All rights reserved. Section 9.5: Critical Points and Optimization Section 9.5 Critical Points and Optimization Functions of More than One Variable

2. Visualizing: Rectangular Coordinate System

3. Visualizing: Points in 3 space

4. Visualizing: Plotting Points in 3 Space

5. Visualizing: Function of 2 variables

6. f(x,y) = y - x 2 Go to Mathematica. Example

7. Applied Calculus, 3/E by Deborah Hughes-Hallet Copyright 2006 by John Wiley & Sons. All rights reserved. Box on page 373

8. Visualizing: Local and Global Extrema

9. Applied Calculus, 3/E by Deborah Hughes-Hallet Copyright 2006 by John Wiley & Sons. All rights reserved. Example: Tables

10. Applied Calculus, 3/E by Deborah Hughes-Hallet Copyright 2006 by John Wiley & Sons. All rights reserved. Example: Contour Diagrams

11. Visualizing: Local and Global Extrema At the local extrema, the first partial derivatives equal zero. At a local maximum (minimum), the second unmixed partial derivatives are usually negative (positive).

12. Applied Calculus, 3/E by Deborah Hughes-Hallet Copyright 2006 by John Wiley & Sons. All rights reserved. Box on page 374 and example 3 Critical Point Analysis

13. Applied Calculus, 3/E by Deborah Hughes-Hallet Copyright 2006 by John Wiley & Sons. All rights reserved. Box on page 374 and example 3 Example: Analytical Computation