Absolute extrema on a closed region
Problem 10.449 · medium
Find the absolute maximum and minimum values of \( \displaystyle f(x, y) = - x^{2} - x y + y^{2} + y \) on the rectangle \( \displaystyle -1 \le x \le 3 \), \( \displaystyle -1 \le y \le 2 \).
- \[ \left[\begin{matrix}\frac{\partial}{\partial x} \left(- x^{2} - x y + y^{2} + y\right)\\\frac{\partial}{\partial y} \left(- x^{2} - x y + y^{2} + y\right)\end{matrix}\right] = \left[\begin{matrix}- 2 x - y\\- x + 2 y + 1\end{matrix}\right] \]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.
- \[ \left[\begin{matrix}-2\\7\\- \frac{1}{5}\\1 \cdot \frac{1}{4}\\-6\\-10\\-9\end{matrix}\right] = \left[\begin{matrix}-2\\7\\- \frac{1}{5}\\\frac{1}{4}\\-6\\-10\\-9\end{matrix}\right] \]f at every candidate: (-1, -1), (-1, 2), (1/5, -2/5), (1/2, -1), (3, -1), (3, 1), (3, 2).✓ Proved
- The largest value is 7, the smallest -10.
Answer \( \max = 7\ \text{at}\ (-1, 2);\ \min = -10\ \text{at}\ (3, 1) \)
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: 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-objectqwen3.6:27b-mlx: inconclusive 2026-10-09 — reviewer response could not be parsed: { "verdict": "fail", "severity": "error", "notes": "The solution lists (1/2, -1) as a candidate point, but this point is not a critical point on any edge nor a corner; it appears to be a calculation egpt-oss:20b: fail (misleading) 2026-10-09 — The solution lists the interior critical point (1/5,‑2/5) but gives an incorrect value for f at that point (the vector of values is wrong). While the final max/min are correct, the erroneous value could mislead a student about how to evaluate f or about the role of interior points.
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.