1. Multivariable Critical Points
Section 10.1
Multivariable Critical Points

2. Idea What type of critical points are there on a 3-D function?
What does the 3-D curve look like at such points?
What does the 2-D contour curve look like at such points?
What does a 2-way table look like at such points?

3. Relative Maximum Point
A relative maximum of a 3-D function is a point that has higher output value (z – value) than any other nearby point. At such a point, the x-cross section and the y-cross section would both have a local maximum.

4. Relative Minimum Point
A relative minimum of a 3-D function is a point that has lower output value (z – value) than any other nearby point. At such a point, the x-cross section and the y-cross section would both have a local minimum.

5. Saddle Point
A saddle point is a point that corresponds to a relative maximum of one cross section but a relative minimum of another cross section.

6. Contour Graphs & Critical Points
A local maximum is located in the center of a series of simple, closed contours that increase in value as we move towards the center.
A local minimum is located in the center of a series of simple, closed contours that decrease in value as we move towards the center.
A saddle point is located in the center of a series of contours that all bend away from the point. In one direction the values increase and in another the values decrease.

7. Contour Curve: Local Max
This is the contour graph for the local maximum point in the earlier slide.
1.7
1.3
1.9
1.5

8. Contour Curve: Local Min
This is the contour graph for the local minimum point in the earlier slide.
-1.5
-1.7
-1.9
-1.3

9. Contour Curve: Saddle Point
This is the contour graph for the saddle point in the earlier slide.
-0.45
-0.25
0.45
0.05
-0.05
0.05
0.45
0.25
-0.05
0.25
-0.25
-0.45

10. Example 1

11. Shaded Contours
Rather than putting the values of the contour curves in the contour graph, sometimes shading is used to indicate larger or smaller values of output (z values). The standard shading has lighter shades corresponding to smaller z values and darker shading corresponding to larger z values.

12. Local Maximum
This is the contour graph for the local maximum point in the earlier slide.
1.7
1.3
1.9
1.5

13. Local Minimum
This is the contour graph for the local minimum point in the earlier slide.
-1.5
-1.7
-1.9
-1.3

14. Saddle Point
This is the contour graph for the saddle point in the earlier slide.
-0.45
-0.25
0.45
0.05
-0.05
0.05
0.45
0.25
-0.05
0.25
-0.25
-0.45

15. Example 2
Find and classify all critical points of the function having the shaded contour plot shown below.

16. Critical Points in Tables
A point in a two-way table is a local maximum point if all 8 cells surrounding it have smaller value.
A point in a two-way table is a local minimum point if all 8 cells surrounding it have larger value.
A point in a two-way table is a saddle point if the values decrease in one direction but increase in the other as you move away from the point.

17. Critical Points in Tables
For example:
Max Min Saddle Saddle

18. Example 3
The table below gives the consistency of cheese spread where x is the percentage of salt and y is the percentage of glycerol used in processing. Find and classify all critical points in the table.

19. Example 4
The table below shows the percentage of sugar converted to olestra when the ratio of peanut oil to sugar is 10:1 at a temperature of yC and a processing time of x hours. Find and classify all critical points in the table.