Absolute extrema on a closed region
Problem 10.508 · medium
Find the absolute maximum and minimum values of \( \displaystyle f(x, y) = x^{2} - x y - 4 x + y^{2} - y \) on the rectangle \( \displaystyle 0 \le x \le 2 \), \( \displaystyle -1 \le y \le 2 \).
- \[ \left[\begin{matrix}\frac{\partial}{\partial x} \left(x^{2} - x y - 4 x + y^{2} - y\right)\\\frac{\partial}{\partial y} \left(x^{2} - x y - 4 x + y^{2} - y\right)\end{matrix}\right] = \left[\begin{matrix}2 x - y - 4\\- x + 2 y - 1\end{matrix}\right] \]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.
- \[ \left[\begin{matrix}2\\- \frac{1}{4}\\2\\- \frac{1}{4}\\0\\- \frac{25}{4}\\-6\end{matrix}\right] \]f at every candidate: (0, -1), (0, 1/2), (0, 2), (3/2, -1), (2, -1), (2, 3/2), (2, 2).✓ Proved
- The largest value is 2, the smallest -25/4.
Answer \( \max = 2\ \text{at}\ (0, -1), (0, 2);\ \min = - \frac{25}{4}\ \text{at}\ (2, \frac{3}{2}) \)
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: passqwen3.6:27b-mlx: fail (error) — The solution fails to identify the interior critical point (3/2, 1/2) and incorrectly lists (0, 1/2) and (3/2, -1) as candidates, which are not on the boundary of the specified rectangle. Consequently, the minimum value is incorrect.
Every verdict on record (4)
gpt-oss:20b: pass 2026-10-11qwen3.6:27b-mlx: fail (error) 2026-10-11 — The solution fails to identify the interior critical point (3/2, 1/2) and incorrectly lists (0, 1/2) and (3/2, -1) as candidates, which are not on the boundary of the specified rectangle. Consequently, the minimum value is incorrect.gpt-oss:20b: pass 2026-10-11qwen3.6:27b-mlx: fail (error) 2026-10-11 — The solution fails to identify the interior critical point (2, 3/2) in the text, listing only boundary points in step 3 despite the answer depending on it. Furthermore, it omits the corner (0, 0) from the candidate list, which is required for a complete analysis of the rectangle's boundary.
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.