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Math 140. 7.3 – Optimizing Functions of Two Variables. has a _________________ at if for all close to . has a _________________ at if for all close to . has a _________________ at if for all close to . has a _________________ at if for all close to .
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Math 140 7.3 – Optimizing Functions of Two Variables
has a _________________at if for all close to . has a _________________at if for all close to .
has a _________________at if for all close to . has a _________________at if for all close to . relative maximum
has a _________________at if for all close to . has a _________________at if for all close to . relative maximum relative minimum
To help find relative extrema, we’ll find _____________, which are points where both and .
To help find relative extrema, we’ll find _____________, which are points where both and . critical points
Relative extremamight happen at critical points, but they might not. See examples below:
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.)
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
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
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
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
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
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
Ex 1. Find all critical points for the function , and classify each as a relative maximum, a relative minimum, or a saddle point.
Ex 2. Find all critical points for the function , and classify each as a relative maximum, a relative minimum, or a saddle point.