Absolute extrema on a closed region
Problem 10.306 · medium
Find the absolute maximum and minimum values of \( \displaystyle f(x, y) = - 2 x^{2} + 3 x + y^{2} - 2 y \) on the rectangle \( \displaystyle 0 \le x \le 2 \), \( \displaystyle 0 \le y \le 1 \).
- \[ \left[\begin{matrix}\frac{\partial}{\partial x} \left(- 2 x^{2} + 3 x + y^{2} - 2 y\right)\\\frac{\partial}{\partial y} \left(- 2 x^{2} + 3 x + y^{2} - 2 y\right)\end{matrix}\right] = \left[\begin{matrix}3 - 4 x\\2 y - 2\end{matrix}\right] \]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.
- \[ \left[\begin{matrix}0\\-1\\\frac{9}{8}\\1 \cdot \frac{1}{8}\\-2\\-3\end{matrix}\right] = \left[\begin{matrix}0\\-1\\\frac{9}{8}\\\frac{1}{8}\\-2\\-3\end{matrix}\right] \]f at every candidate: (0, 0), (0, 1), (3/4, 0), (3/4, 1), (2, 0), (2, 1).✓ Proved
- The largest value is 9/8, the smallest -3.
Answer \( \max = \frac{9}{8}\ \text{at}\ (\frac{3}{4}, 0);\ \min = -3\ \text{at}\ (2, 1) \)
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: fail (misleading) — The solution omits the interior critical point (3/4,1) and incorrectly reports its function value as 9/8 instead of 1/8, which could mislead a student about the correct evaluation of f at that point.qwen3.6:27b-mlx: fail (style) — [domain objection, downgraded to style] The solution fails to identify the interior critical point (3/4, 1), which lies within the domain and yields a local minimum of -17/8. Although this value 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)
qwen3.6:27b-mlx: fail (style) 2026-10-04 — [domain objection, downgraded to style] The solution fails to identify the interior critical point (3/4, 1), which lies within the domain and yields a local minimum of -17/8. Although this value 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: fail (misleading) 2026-10-04 — The solution omits the interior critical point (3/4,1) and incorrectly reports its function value as 9/8 instead of 1/8, which could mislead a student about the correct evaluation of f at that point.qwen3.6:27b-mlx: fail (style) 2026-10-04 — [domain objection, downgraded to style] The solution fails to identify the interior critical point (3/4, 1), which lies within the domain and yields a local minimum of -17/8. While this is not the global minimum, omitting it is a procedural error in finding all critical points.gpt-oss:20b: pass 2026-10-04
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 2026-10-04 with SymPy 1.14.0.