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Chapter 12: Q. 7 (page 987)
Give precise mathematical definitions or descriptions of each of the following concepts that follow. Then illustrate the definition with a graph or an algebraic example.
* a closed subset of
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Let be a differentiable function such that for every point in the domain of f, and let be a closed, bounded subset of role="math" localid="1649887954022" Explain why the maximum and minimum of f restricted to occur on the boundary ofrole="math" localid="1649888770915"
When you use the method of Lagrange multipliers to find the maximum and minimum of subject to the constraint you obtain two points. Is there a relative maximum at one of the points and a relative minimum at the other? Which is which?
In Example 4 we found that the function has stationary points at and
(a) Use the second-derivative test to show that \(f\) has a saddle point at
(b) Use the second-derivative test to show that \(f\) has a relative minimum at
(c) Use the value of \(f(-10,0)\) to argue that \(f\) has a relative minimum at and not an absolute minimum, without using the second-derivative test.
Partial derivatives: Find all first- and second-order partial derivatives for the following functions:
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