Questions: Optimization in Multiple Variables

Question 1

You are maximizing a continuous function f(x, y) on a closed bounded region D. You find two interior critical points and also optimize f along the boundary. Where must the global maximum occur?

• A. At one of the interior critical points where ∇f = 0
• B. On the boundary of D
• C. At either a critical point or on the boundary — you must compare all candidates
• D. At the critical point where the Hessian is negative definite

Answer: (C)

The Extreme Value Theorem guarantees a global max exists on a closed bounded set, but it can occur at an interior critical point OR on the boundary. You must evaluate f at all critical points and all boundary candidates, then compare every value. Stopping after finding interior critical points, or assuming the negative-definite Hessian point is the global max, is the most common error on multivariable optimization problems.

Question 2

If a differentiable function f(x, y) has exactly one critical point in its domain and the Hessian at that point is positive definite (confirming a local minimum), then that point should be the global minimum of f.

• T. True
• F. False

Answer: False

This is only guaranteed if the domain is closed and bounded (by the Extreme Value Theorem) and you have also checked the boundary. On an unbounded domain — or even on a closed bounded region where the boundary has not been checked — f could attain smaller values outside the interior. A local minimum is only a global minimum if no other point (including boundary points) yields a smaller value.

Question 3

Describe the complete procedure for finding the global maximum and minimum of a continuous function f(x, y) on a closed bounded region D.

Model answer: Step 1: Find all interior critical points by solving ∇f = 0 and evaluate f at each. Step 2: Parameterize each piece of the boundary and optimize f restricted to that piece (a single-variable problem). Step 3: Collect all candidate values from steps 1 and 2. Step 4: The global max is the largest value; the global min is the smallest.

The Extreme Value Theorem guarantees both extrema exist on a closed bounded set, so this exhaustive comparison always terminates with valid answers. Steps 1 and 2 together cover all possible locations: the interior (where ∇f = 0 at any interior extremum) and the boundary (where the extremum need not satisfy ∇f = 0). Stopping after step 1 is a very common mistake — the global extremum often lies on the boundary.