I'm working on creating a calculus exercise involving a function of two variables and am looking for suggestions to meet some specific criteria. The exercise is designed for students to practice finding global extrema, both on the interior and along the boundary of a defined domain. However, I'm encountering difficulties in defining a function and a suitable domain that satisfy all of the following requirements:
Function Characteristics: The function should have second partial derivatives that depend on x and/or y. This requirement is to remind students to evaluate the second order of partial derivatives at critical points when using the second-order derivative test.
Critical Points: The function must have at least one critical point within the interior of the domain (not on the boundary).
Domain: The domain should be a closed area bounded by simple functions or lines. Key requirements for the domain and the positioning of the global extrema include:
- Non-Corner Boundary Extrema Requirement for One Extremum: Either the global maximum or the global minimum (not necessarily both) should occur along a boundary segment, but not at the vertices where two boundary lines or curves meet. This extremum should be a "true" boundary point, distinct from corners/vertices.
- One-Variable Function Transformation Along Boundary Segments: When examining a boundary line segment, students should be able to transform the multivariable function into a one-variable function along that segment. The critical task is to find the critical points of this one-variable function by differentiating it and setting the derivative to zero. One of these critical points, located along the boundary segments (and not at the endpoints of these segments), should correspond to either the global maximum or minimum (as per the above criterion).
Criteria for Global Extrema:
Interior Critical Point: Ideally, either the global maximum or the global minimum should occur at an interior critical point, where the gradient of the function vanishes. While this is a preferred condition, it is not an absolute requirement.
Non-Corner Boundary Extrema: It is essential that either the global maximum or the global minimum occurs at a true boundary point (not at vertices where two boundary lines intersect). This means one of the global extrema should be found along the boundary segments, distinctly away from any corner points (vertices of line segments) of the domain.
I've attempted several functions and domains, but each time, the global maximum or minimum ends up being at a corner of the boundary, or there are no suitable interior critical points.
Could anyone suggest a function and a corresponding domain that meets these criteria? The solution should be feasible to solve by hand, as this is intended for a calculus exercise.
Any insights or examples would be greatly appreciated!