Absolute extrema on a closed region
Problem 10.453 · medium
Find the absolute maximum and minimum values of \( \displaystyle f(x, y) = - 2 x^{2} + x y - 2 x + 2 y^{2} + 3 y \) on the rectangle \( \displaystyle -2 \le x \le 3 \), \( \displaystyle 0 \le y \le 3 \).
- \[ \left[\begin{matrix}\frac{\partial}{\partial x} \left(- 2 x^{2} + x y - 2 x + 2 y^{2} + 3 y\right)\\\frac{\partial}{\partial y} \left(- 2 x^{2} + x y - 2 x + 2 y^{2} + 3 y\right)\end{matrix}\right] = \left[\begin{matrix}- 4 x + y - 2\\x + 4 y + 3\end{matrix}\right] \]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.
- \[ \left[\begin{matrix}-4\\17\\1 \cdot \frac{1}{2}\\\frac{217}{8}\\-24\\12\end{matrix}\right] = \left[\begin{matrix}-4\\17\\\frac{1}{2}\\\frac{217}{8}\\-24\\12\end{matrix}\right] \]f at every candidate: (-2, 0), (-2, 3), (-1/2, 0), (1/4, 3), (3, 0), (3, 3).✓ Proved
- The largest value is 217/8, the smallest -24.
Answer \( \max = \frac{217}{8}\ \text{at}\ (\frac{1}{4}, 3);\ \min = -24\ \text{at}\ (3, 0) \)
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: passqwen3.6:27b-mlx: inconclusive — reviewer returned a non-object
Every verdict on record (4)
gpt-oss:20b: pass 2026-10-09qwen3.6:27b-mlx: inconclusive 2026-10-09 — reviewer returned a non-objectgpt-oss:20b: fail (style) 2026-10-09 — [domain objection, downgraded to style] The minimum value is incorrectly reported. f(3,0) = -18, not -24, and no other point in the rectangle attains -24. The solution also lists an interior point (-1/2,0) that lies on the boundary, but the interior critical point actually lies outside the domain. Thus the conclusion about the absolute extrema is false.qwen3.6:27b-mlx: fail (error) 2026-10-09 — The solution fails to identify the interior critical point (-1/2, 3/8) and incorrectly lists (-1/2, 0) as a candidate, which is not a critical point on the boundary y=0. Consequently, the global minimum is likely incorrect.
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 2026-10-09 with SymPy 1.14.0.