Absolute extrema on a closed region
Problem 10.563 · medium
Find the absolute maximum and minimum values of \( \displaystyle f(x, y) = x^{2} + x y - 4 x - 2 y^{2} + y \) on the rectangle \( \displaystyle -1 \le x \le 2 \), \( \displaystyle -2 \le y \le 3 \).
- \[ \left[\begin{matrix}\frac{\partial}{\partial x} \left(x^{2} + x y - 4 x - 2 y^{2} + y\right)\\\frac{\partial}{\partial y} \left(x^{2} + x y - 4 x - 2 y^{2} + y\right)\end{matrix}\right] = \left[\begin{matrix}2 x + y - 4\\x - 4 y + 1\end{matrix}\right] \]Interior critical points solve ∇f = 0.✓ Proved
- On each edge f is a function of one variable: find its critical points there too, and include the four corners.
- \[ \left[\begin{matrix}-3\\5\\-13\\- \frac{61}{4}\\-3\\-18\\- \frac{23}{8}\\-13\end{matrix}\right] \]f at every candidate: (-1, -2), (-1, 0), (-1, 3), (1/2, 3), (5/3, 2/3), (2, -2), (2, 3/4), (2, 3).✓ Proved
- The largest value is 5, the smallest -18.
Answer \( \max = 5\ \text{at}\ (-1, 0);\ \min = -18\ \text{at}\ (2, -2) \)
Lines: 2 proved, 2 not checked. The answer was also checked a second way, without looking at the solution. The reviewers disagree about how one step is explained; every verdict is in the receipt.
The full receipt
| Line | Status | Checked by | Detail |
|---|---|---|---|
| 1 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| 2 | Not checked | — | a sentence; read, not computed |
| 3 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| 4 | Not checked | — | a sentence; read, not computed |
| answer, a second way | ✓ Checked independently | sympy 1.14.0 + mpmath 1.3.0 | f sampled on a 241 × 241 grid never beats the claimed max or min, and comes within 5% of both |
Reviewers
gpt-oss:20b: fail (error) — The computed values at several candidate points are incorrect. For example, f(-1,3) = -21, f(1/2,3) ≈ -15.25, and f(5/3,2/3) = -3, yet the solution lists -13 for all three. These errors invalidate the conclusion that the minimum is -18 at (2,-2) and the maximum is 5 at (-1,0).qwen3.6:27b-mlx: pass — The solution correctly identifies the interior critical point and the boundary candidates, evaluates the function at all relevant points, and correctly identifies the global maximum and minimum.
Every verdict on record (4)
gpt-oss:20b: fail (error) 2026-10-13 — The computed values at several candidate points are incorrect. For example, f(-1,3) = -21, f(1/2,3) ≈ -15.25, and f(5/3,2/3) = -3, yet the solution lists -13 for all three. These errors invalidate the conclusion that the minimum is -18 at (2,-2) and the maximum is 5 at (-1,0).qwen3.6:27b-mlx: pass 2026-10-13 — The solution correctly identifies the interior critical point and the boundary candidates, evaluates the function at all relevant points, and correctly identifies the global maximum and minimum.gpt-oss:20b: pass 2026-10-13qwen3.6:27b-mlx: fail (error) 2026-10-13 — The solution fails to evaluate the function at the interior critical point (5/3, 2/3), which is a required step for finding absolute extrema on a closed region. Although the final answer happens to be correct because the boundary extrema are more extreme, omitting the interior check is a methodological error that would lead to incorrect results in many similar problems.
Proved: SymPy reduced the difference between the two sides to zero. Checked independently: a separate
method, named above, confirmed it. Checked numerically: the two sides agree at every sampled point, which is evidence,
not proof. Reviewed: a model or a person read it; that is all a sentence can have. Solution by
generator:structured/absolute_extrema_2var, checked 2026-10-13 with SymPy 1.14.0.