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Absolute extrema on a closed region

Problem 10.504 · medium

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

\[ \left[\begin{matrix}\frac{\partial}{\partial x} \left(- 2 x^{2} + x y - x + y^{2} + y\right)\\\frac{\partial}{\partial y} \left(- 2 x^{2} + x y - x + y^{2} + y\right)\end{matrix}\right] = \left[\begin{matrix}- 4 x + y - 1\\x + 2 y + 1\end{matrix}\right] \]
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.
\[ \left[\begin{matrix}0\\- \frac{25}{4}\\0\\\frac{25}{8}\\0\\\frac{25}{2}\\-25\\0\end{matrix}\right] \]
f at every candidate: (-2, -2), (-2, 1/2), (-2, 3), (-3/4, -2), (-1/3, -1/3), (1/2, 3), (3, -2), (3, 3).✓ Proved
The largest value is 25/2, the smallest -25.

Answer \( \max = \frac{25}{2}\ \text{at}\ (\frac{1}{2}, 3);\ \min = -25\ \text{at}\ (3, -2) \)

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 (error) — The solution claims to evaluate f at 'every candidate' but omits the interior critical point (-1/3, -1/3) from the list of points used to determine the extrema, although its value (25/8) is present in the matrix. More critically, it fails to explicitly check the other boundary segments (x=3 and y=-2) for interior critical points, only listing corners and points on x=-2 and y=3. While the final answer happens to be correct because the global extrema are on the boundary, the method described is incomplete and misleading as it implies a systematic check was performed when it was not.

Every verdict on record (4)

gpt-oss:20b: pass 2026-10-11
qwen3.6:27b-mlx: fail (error) 2026-10-11 — The solution claims to evaluate f at 'every candidate' but omits the interior critical point (-1/3, -1/3) from the list of points used to determine the extrema, although its value (25/8) is present in the matrix. More critically, it fails to explicitly check the other boundary segments (x=3 and y=-2) for interior critical points, only listing corners and points on x=-2 and y=3. While the final answer happens to be correct because the global extrema are on the boundary, the method described is incomplete and misleading as it implies a systematic check was performed when it was not.
gpt-oss:20b: pass 2026-10-11
qwen3.6:27b-mlx: fail (error) 2026-10-11 — The solution fails to find the interior critical point by solving the system of equations in step 1, omitting the candidate (-1/3, -1/3) from the logical derivation. Although this point appears in the list in step 3, the solution does not show how it was obtained, making the method incomplete and misleading.

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 2026-10-11 with SymPy 1.14.0.