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Absolute extrema on a closed region

Problem 10.355 · medium

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

\[ \left[\begin{matrix}\frac{\partial}{\partial x} \left(x^{2} + x y - x + y^{2} + 4 y\right)\\\frac{\partial}{\partial y} \left(x^{2} + x y - x + y^{2} + 4 y\right)\end{matrix}\right] = \left[\begin{matrix}2 x + y - 1\\x + 2 y + 4\end{matrix}\right] \]
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.
\[ \left[\begin{matrix}0\\- \frac{1}{4}\\6\\5\\- \frac{25}{4}\\-4\\14\end{matrix}\right] \]
f at every candidate: (-1, -2), (-1, -3/2), (-1, 1), (0, 1), (3/2, -2), (3, -2), (3, 1).✓ Proved
The largest value is 14, the smallest -25/4.

Answer \( \max = 14\ \text{at}\ (3, 1);\ \min = - \frac{25}{4}\ \text{at}\ (\frac{3}{2}, -2) \)

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 lists (-1, -3/2) as a candidate point, but y = -3/2 is not on the boundary of the rectangle (y ranges from -2 to 1); the critical point on the edge x=-1 is actually at y=-1/4. This indicates a calculation error in finding the boundary critical points.

Every verdict on record (4)

qwen3.6:27b-mlx: fail (error) 2026-10-05 — The solution lists (-1, -3/2) as a candidate point, but y = -3/2 is not on the boundary of the rectangle (y ranges from -2 to 1); the critical point on the edge x=-1 is actually at y=-1/4. This indicates a calculation error in finding the boundary critical points.
gpt-oss:20b: pass 2026-10-05
qwen3.6:27b-mlx: inconclusive 2026-10-05 — reviewer returned a non-object
gpt-oss:20b: fail (error) 2026-10-05 — The interior critical point is incorrectly identified. Solving ∇f=0 gives (x,y)=(2,-3), which lies outside the rectangle, so there is no interior critical point. The solution lists (0,1) as a candidate, but (0,1) is not on the boundary of the rectangle. Consequently the set of candidates is incomplete and the conclusion about the extrema is not justified.

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 2026-10-05 with SymPy 1.14.0.