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Absolute extrema on a closed region

Problem 10.561 · medium

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

\[ \left[\begin{matrix}\frac{\partial}{\partial x} \left(- 2 x^{2} - x y + 2 x - y^{2} + 4 y\right)\\\frac{\partial}{\partial y} \left(- 2 x^{2} - x y + 2 x - y^{2} + 4 y\right)\end{matrix}\right] = \left[\begin{matrix}- 4 x - y + 2\\- x - 2 y + 4\end{matrix}\right] \]
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.
\[ \left[\begin{matrix}-18\\0\\\frac{25}{8}\\-10\\2\end{matrix}\right] \]
f at every candidate: (-1, -2), (-1, 1), (1/4, 1), (1, -2), (1, 1).✓ Proved
The largest value is 25/8, the smallest -18.

Answer \( \max = \frac{25}{8}\ \text{at}\ (\frac{1}{4}, 1);\ \min = -18\ \text{at}\ (-1, -2) \)

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 (error) — The solution fails to explicitly calculate or list the critical points on the boundary edges (where the derivative with respect to the varying coordinate is zero), only listing corners and the interior point. While the final answer happens to be correct because the extrema occur at the interior point and a corner, the method described in step 2 is not executed, making the solution incomplete and potentially misleading about the procedure.

Every verdict on record (4)

gpt-oss:20b: pass 2026-10-13
qwen3.6:27b-mlx: fail (error) 2026-10-13 — The solution fails to explicitly calculate or list the critical points on the boundary edges (where the derivative with respect to the varying coordinate is zero), only listing corners and the interior point. While the final answer happens to be correct because the extrema occur at the interior point and a corner, the method described in step 2 is not executed, making the solution incomplete and potentially misleading about the procedure.
gpt-oss:20b: pass 2026-10-13
qwen3.6:27b-mlx: fail (error) 2026-10-13 — The solution fails to explicitly calculate or list the critical points on the boundary edges (where the derivative with respect to the free variable is zero), only listing corners and the interior point. While the final answer happens to be correct because the extrema occur at a corner and the interior point, the method described is incomplete and misleading as it omits a necessary step for finding boundary extrema.

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.