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Absolute extrema on a closed region

Problem 10.402 · medium

Find the absolute maximum and minimum values of \( \displaystyle f(x, y) = x^{2} - x y + 4 x + y^{2} \) on the rectangle \( \displaystyle -1 \le x \le 3 \), \( \displaystyle 0 \le y \le 1 \).

\[ \left[\begin{matrix}\frac{\partial}{\partial x} \left(x^{2} - x y + 4 x + y^{2}\right)\\\frac{\partial}{\partial y} \left(x^{2} - x y + 4 x + y^{2}\right)\end{matrix}\right] = \left[\begin{matrix}2 x - y + 4\\- x + 2 y\end{matrix}\right] \]
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.
\[ \left[\begin{matrix}-3\\-1\\21\\19\end{matrix}\right] \]
f at every candidate: (-1, 0), (-1, 1), (3, 0), (3, 1).✓ Proved
The largest value is 21, the smallest -3.

Answer \( \max = 21\ \text{at}\ (3, 0);\ \min = -3\ \text{at}\ (-1, 0) \)

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 for interior critical points. Solving ∇f = 0 yields the point (-4, -2), which is outside the domain, but this step must be explicitly performed or stated to justify relying solely on the boundary. Furthermore, the solution ignores critical points on the edges (e.g., on y=0, f_x=0 at x=-2, outside; on y=1, f_x=0 at x=-1.5, outside; on x=-1, f_y=0 at y=-0.5, outside; on x=3, f_y=0 at y=1.5, outside). While the corners happen to be the extrema, the method described in sentence 2 implies checking edge critical points, but sentence 3 only lists corners, omitting the verification that no edge critical points exist within the segments.

Every verdict on record (4)

gpt-oss:20b: pass 2026-10-07
qwen3.6:27b-mlx: fail (style) 2026-10-07 — [domain objection, downgraded to style] The solution fails to check for interior critical points. Solving ∇f = 0 yields the point (-4, -2), which is outside the domain, but this step must be explicitly performed or stated to justify relying solely on the boundary. Furthermore, the solution ignores critical points on the edges (e.g., on y=0, f_x=0 at x=-2, outside; on y=1, f_x=0 at x=-1.5, outside; on x=-1, f_y=0 at y=-0.5, outside; on x=3, f_y=0 at y=1.5, outside). While the corners happen to be the extrema, the method described in sentence 2 implies checking edge critical points, but sentence 3 only lists corners, omitting the verification that no edge critical points exist within the segments.
qwen3.6:27b-mlx: fail (error) 2026-10-07 — The solution fails to find the interior critical point (x=-4, y=-2) and incorrectly claims that checking only the corners is sufficient for the boundary analysis, ignoring potential extrema on the edges themselves.
gpt-oss:20b: pass 2026-10-07

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 2026-10-07 with SymPy 1.14.0.