Absolute extrema on a closed region
Problem 10.450 · medium
Find the absolute maximum and minimum values of \( \displaystyle f(x, y) = - 2 x^{2} + 2 x - y^{2} + 4 y \) on the rectangle \( \displaystyle -2 \le x \le 2 \), \( \displaystyle -1 \le y \le 3 \).
- \[ \left[\begin{matrix}\frac{\partial}{\partial x} \left(- 2 x^{2} + 2 x - y^{2} + 4 y\right)\\\frac{\partial}{\partial y} \left(- 2 x^{2} + 2 x - y^{2} + 4 y\right)\end{matrix}\right] = \left[\begin{matrix}2 - 4 x\\4 - 2 y\end{matrix}\right] \]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.
- \[ \left[\begin{matrix}-17\\-8\\-9\\- \frac{9}{2}\\\frac{9}{2}\\\frac{7}{2}\\-9\\0\\-1\end{matrix}\right] \]f at every candidate: (-2, -1), (-2, 2), (-2, 3), (1/2, -1), (1/2, 2), (1/2, 3), (2, -1), (2, 2), (2, 3).✓ Proved
- The largest value is 9/2, the smallest -17.
Answer \( \max = \frac{9}{2}\ \text{at}\ (\frac{1}{2}, 2);\ \min = -17\ \text{at}\ (-2, -1) \)
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 on the edge x = –2 (at (–2, 2)), which should be evaluated. While it does not affect the final extrema, the omission could mislead a student into thinking all edge candidates were considered.qwen3.6:27b-mlx: pass — The solution correctly identifies the interior critical point, evaluates the function at all relevant candidates (interior critical points, boundary critical points, and corners), and correctly determines the absolute maximum and minimum values.
Every verdict on record (4)
gpt-oss:20b: fail (misleading) 2026-10-09 — The solution omits the interior critical point on the edge x = –2 (at (–2, 2)), which should be evaluated. While it does not affect the final extrema, the omission could mislead a student into thinking all edge candidates were considered.qwen3.6:27b-mlx: pass 2026-10-09 — The solution correctly identifies the interior critical point, evaluates the function at all relevant candidates (interior critical points, boundary critical points, and corners), and correctly determines the absolute maximum and minimum values.qwen3.6:27b-mlx: pass 2026-10-09 — The solution correctly identifies the interior critical point and evaluates the function at all relevant boundary candidates (corners and edge critical points). The final comparison yields the correct absolute maximum and minimum values.gpt-oss:20b: pass 2026-10-09
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.