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Absolute extrema on a closed region

Problem 10.304 · medium

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

\[ \left[\begin{matrix}\frac{\partial}{\partial x} \left(- x^{2} + 2 x - 2 y^{2}\right)\\\frac{\partial}{\partial y} \left(- x^{2} + 2 x - 2 y^{2}\right)\end{matrix}\right] = \left[\begin{matrix}2 - 2 x\\- 4 y\end{matrix}\right] \]
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.
Reviewed
\[ \left[\begin{matrix}0\\-2\\1\\-1\\-3\\-5\end{matrix}\right] \]
f at every candidate: (0, 0), (0, 1), (1, 0), (1, 1), (3, 0), (3, 1).✓ Proved
The largest value is 1, the smallest -5.
Reviewed

Answer \( \max = 1\ \text{at}\ (1, 0);\ \min = -5\ \text{at}\ (3, 1) \)

✓ Nihil obstat Lines: 2 proved, 2 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation.

The full receipt

Line	Status	Checked by	Detail
1	✓ Proved	sympy 1.14.0	simplify(lhs - rhs) reduced to 0
2	Reviewed	reviewers: gpt-oss:20b, qwen3.6:27b-mlx	a sentence; read, not computed
3	✓ Proved	sympy 1.14.0	simplify(lhs - rhs) reduced to 0
4	Reviewed	reviewers: gpt-oss:20b, qwen3.6:27b-mlx	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: pass — The solution correctly identifies the interior critical point and evaluates the function at all relevant candidates (interior critical points, boundary critical points, and corners). The final comparison yields the correct absolute maximum and minimum.

Every verdict on record (4)

qwen3.6:27b-mlx: pass 2026-10-04 — The solution correctly identifies the interior critical point and evaluates the function at all relevant candidates (interior critical points, boundary critical points, and corners). The final comparison yields the correct absolute maximum and minimum.
gpt-oss:20b: pass 2026-10-04
qwen3.6:27b-mlx: pass 2026-10-04 — The solution correctly identifies the interior critical point and evaluates the function at all necessary candidates (interior critical points, boundary critical points, and corners). The final comparison of values is correct.
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.