Absolute extrema on a closed region
Problem 10.301 · medium
Find the absolute maximum and minimum values of \( \displaystyle f(x, y) = x^{2} + x y - 4 x - 2 y^{2} + 4 y \) on the rectangle \( \displaystyle -1 \le x \le 3 \), \( \displaystyle -2 \le y \le 2 \).
- \[ \left[\begin{matrix}\frac{\partial}{\partial x} \left(x^{2} + x y - 4 x - 2 y^{2} + 4 y\right)\\\frac{\partial}{\partial y} \left(x^{2} + x y - 4 x - 2 y^{2} + 4 y\right)\end{matrix}\right] = \left[\begin{matrix}2 x + y - 4\\x - 4 y + 4\end{matrix}\right] \]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.
- \[ \left[\begin{matrix}-9\\\frac{49}{8}\\3\\-1\\0\\-25\\\frac{25}{8}\\3\end{matrix}\right] \]f at every candidate: (-1, -2), (-1, 3/4), (-1, 2), (1, 2), (4/3, 4/3), (3, -2), (3, 7/4), (3, 2).✓ Proved
- The largest value is 49/8, the smallest -25.
Answer \( \max = \frac{49}{8}\ \text{at}\ (-1, \frac{3}{4});\ \min = -25\ \text{at}\ (3, -2) \)
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: inconclusive — reviewer returned a non-object
Every verdict on record (4)
qwen3.6:27b-mlx: inconclusive 2026-10-04 — reviewer returned a non-objectgpt-oss:20b: pass 2026-10-04qwen3.6:27b-mlx: fail (error) 2026-10-04 — The solution lists (4/3, 4/3) as a candidate point, but this point is not a critical point of f (the actual interior critical point is (4/3, 7/4)). Furthermore, the value 25/8 listed for (4/3, 4/3) is incorrect; f(4/3, 4/3) = 1/3. The correct interior critical point (4/3, 7/4) yields f = 49/8, which is already captured by the boundary point (-1, 3/4) in the list, but the reasoning is flawed and contains calculation errors.gpt-oss:20b: fail (error) 2026-10-04 — The solution lists candidate points that are not on the boundary of the rectangle (e.g., (1,2) and (3,7/4)) and omits the correct boundary critical points. The edge analysis is incomplete and incorrectly enumerated, leading to an unreliable determination of the extrema.
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.