Absolute extrema on a closed region
Problem 10.299 · medium
Find the absolute maximum and minimum values of \( \displaystyle f(x, y) = - 2 x^{2} - x y - x - y^{2} - 2 y \) on the rectangle \( \displaystyle -1 \le x \le 3 \), \( \displaystyle -2 \le y \le 3 \).
- \[ \left[\begin{matrix}\frac{\partial}{\partial x} \left(- 2 x^{2} - x y - x - y^{2} - 2 y\right)\\\frac{\partial}{\partial y} \left(- 2 x^{2} - x y - x - y^{2} - 2 y\right)\end{matrix}\right] = \left[\begin{matrix}- 4 x - y - 1\\- x - 2 y - 2\end{matrix}\right] \]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.
- \[ \left[\begin{matrix}-3\\- \frac{3}{4}\\-13\\1\\1 \cdot \frac{1}{8}\\-15\\-45\end{matrix}\right] = \left[\begin{matrix}-3\\- \frac{3}{4}\\-13\\1\\\frac{1}{8}\\-15\\-45\end{matrix}\right] \]f at every candidate: (-1, -2), (-1, -1/2), (-1, 3), (0, -1), (1/4, -2), (3, -2), (3, 3).✓ Proved
- The largest value is 1, the smallest -45.
Answer \( \max = 1\ \text{at}\ (0, -1);\ \min = -45\ \text{at}\ (3, 3) \)
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 (misleading) — The value of f at (-1,3) is computed incorrectly as -13; the correct value is -19. This misstatement could mislead a student about the function’s behavior on the boundary, even though the final extrema remain correct.qwen3.6:27b-mlx: inconclusive — reviewer returned a non-object
Every verdict on record (4)
qwen3.6:27b-mlx: inconclusive 2026-10-04 — reviewer returned a non-objectgpt-oss:20b: fail (misleading) 2026-10-04 — The value of f at (-1,3) is computed incorrectly as -13; the correct value is -19. This misstatement could mislead a student about the function’s behavior on the boundary, even though the final extrema remain correct.qwen3.6:27b-mlx: pass 2026-10-04 — 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 2026-10-04
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 2026-10-04 with SymPy 1.14.0.