Math 140

1. Math 140 7.3 – Optimizing Functions of Two Variables

2. has a _________________at if for all close to . has a _________________at if for all close to .

3. has a _________________at if for all close to . has a _________________at if for all close to . relative maximum

4. has a _________________at if for all close to . has a _________________at if for all close to . relative maximum relative minimum

5. To help find relative extrema, we’ll find _____________, which are points where both and .

6. To help find relative extrema, we’ll find _____________, which are points where both and . critical points

7. Relative extremamight happen at critical points, but they might not. See examples below:

8. The Second Partials Test Let 1. Find . 2. Find all ______________of . 3. For each critical point , evaluate __________. 4. If , then there is a ____________at . 5. If , then compute . a. If , then there is a ____________at . b. If , then there is a ____________at . (Note: If , then test doesn’t tell you anything.)

9. The Second Partials Test Let 1. Find . 2. Find all ______________of . 3. For each critical point , evaluate __________. 4. If , then there is a ____________at . 5. If , then compute . a. If , then there is a ____________at . b. If , then there is a ____________at . (Note: If , then test doesn’t tell you anything.) critical points

10. The Second Partials Test Let 1. Find . 2. Find all ______________of . 3. For each critical point , evaluate __________. 4. If , then there is a ____________at . 5. If , then compute . a. If , then there is a ____________at . b. If , then there is a ____________at . (Note: If , then test doesn’t tell you anything.) critical points

11. The Second Partials Test Let 1. Find . 2. Find all ______________of . 3. For each critical point , evaluate __________. 4. If , then there is a ____________at . 5. If , then compute . a. If , then there is a ____________at . b. If , then there is a ____________at . (Note: If , then test doesn’t tell you anything.) critical points saddle point

12. The Second Partials Test Let 1. Find . 2. Find all ______________of . 3. For each critical point , evaluate __________. 4. If , then there is a ____________at . 5. If , then compute . a. If , then there is a ____________at . b. If , then there is a ____________at . (Note: If , then test doesn’t tell you anything.) critical points saddle point relative min

13. The Second Partials Test Let 1. Find . 2. Find all ______________of . 3. For each critical point , evaluate __________. 4. If , then there is a ____________at . 5. If , then compute . a. If , then there is a ____________at . b. If , then there is a ____________at . (Note: If , then test doesn’t tell you anything.) critical points saddle point relative min relative max

14. The Second Partials Test Let 1. Find . 2. Find all ______________of . 3. For each critical point , evaluate __________. 4. If , then there is a ____________at . 5. If , then compute . a. If , then there is a ____________at . b. If , then there is a ____________at . (Note: If , then test doesn’t tell you anything.) critical points saddle point relative min relative max

15. Ex 1. Find all critical points for the function , and classify each as a relative maximum, a relative minimum, or a saddle point.

16. Ex 2. Find all critical points for the function , and classify each as a relative maximum, a relative minimum, or a saddle point.