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Absolute extrema on a closed region

Problem 10.557 · medium

Find the absolute maximum and minimum values of \( \displaystyle f(x, y) = x^{2} - 2 x - y^{2} + 3 y \) on the rectangle \( \displaystyle 0 \le x \le 1 \), \( \displaystyle -2 \le y \le 1 \).

\[ \left[\begin{matrix}\frac{\partial}{\partial x} \left(x^{2} - 2 x - y^{2} + 3 y\right)\\\frac{\partial}{\partial y} \left(x^{2} - 2 x - y^{2} + 3 y\right)\end{matrix}\right] = \left[\begin{matrix}2 x - 2\\3 - 2 y\end{matrix}\right] \]
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.
\[ \left[\begin{matrix}-10\\2\\-11\\1\end{matrix}\right] \]
f at every candidate: (0, -2), (0, 1), (1, -2), (1, 1).✓ Proved
The largest value is 2, the smallest -11.

Answer \( \max = 2\ \text{at}\ (0, 1);\ \min = -11\ \text{at}\ (1, -2) \)

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: pass
qwen3.6:27b-mlx: fail (style) — [domain objection, downgraded to style] The solution fails to check the interior critical point (1, 3/2), which lies within the domain and yields a local maximum of 3.5. By omitting this point, the solution incorrectly identifies 2 as the absolute maximum.

Every verdict on record (4)

gpt-oss:20b: pass 2026-10-13
qwen3.6:27b-mlx: fail (style) 2026-10-13 — [domain objection, downgraded to style] The solution fails to check the interior critical point (1, 3/2), which lies within the domain and yields a local maximum of 3.5. By omitting this point, the solution incorrectly identifies 2 as the absolute maximum.
qwen3.6:27b-mlx: inconclusive 2026-10-13 — reviewer returned a non-object
gpt-oss:20b: pass 2026-10-13

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.