Absolute extrema on a closed region
Problem 10.562 · medium
Find the absolute maximum and minimum values of \( \displaystyle f(x, y) = 2 x^{2} + x y - 2 x - 2 y^{2} - y \) on the rectangle \( \displaystyle -2 \le x \le 3 \), \( \displaystyle 0 \le y \le 1 \).
- \[ \left[\begin{matrix}\frac{\partial}{\partial x} \left(2 x^{2} + x y - 2 x - 2 y^{2} - y\right)\\\frac{\partial}{\partial y} \left(2 x^{2} + x y - 2 x - 2 y^{2} - y\right)\end{matrix}\right] = \left[\begin{matrix}4 x + y - 2\\x - 4 y - 1\end{matrix}\right] \]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.
- \[ \left[\begin{matrix}12\\7\\- \frac{25}{8}\\- \frac{1}{2}\\12\\\frac{25}{2}\\12\end{matrix}\right] \]f at every candidate: (-2, 0), (-2, 1), (1/4, 1), (1/2, 0), (3, 0), (3, 1/2), (3, 1).✓ Proved
- The largest value is 25/2, the smallest -25/8.
Answer \( \max = \frac{25}{2}\ \text{at}\ (3, \frac{1}{2});\ \min = - \frac{25}{8}\ \text{at}\ (\frac{1}{4}, 1) \)
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 (style) — [domain objection, downgraded to style] The solution fails to evaluate the function at the interior critical point (1/4, -1/4), which lies within the domain. Although this point is not the global minimum, omitting it from the candidate list is a procedural error in finding extrema on a closed region.
Every verdict on record (4)
gpt-oss:20b: pass 2026-10-13qwen3.6:27b-mlx: fail (style) 2026-10-13 — [domain objection, downgraded to style] The solution fails to evaluate the function at the interior critical point (1/4, -1/4), which lies within the domain. Although this point is not the global minimum, omitting it from the candidate list is a procedural error in finding extrema on a closed region.gpt-oss:20b: pass 2026-10-13qwen3.6:27b-mlx: inconclusive 2026-10-13 — reviewer returned a non-object
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.