Absolute extrema on a closed region
Problem 10.400 · medium
Find the absolute maximum and minimum values of \( \displaystyle f(x, y) = 2 x^{2} + x y + 3 x - 2 y^{2} - 4 y \) on the rectangle \( \displaystyle -2 \le x \le 3 \), \( \displaystyle -2 \le y \le 3 \).
- \[ \left[\begin{matrix}\frac{\partial}{\partial x} \left(2 x^{2} + x y + 3 x - 2 y^{2} - 4 y\right)\\\frac{\partial}{\partial y} \left(2 x^{2} + x y + 3 x - 2 y^{2} - 4 y\right)\end{matrix}\right] = \left[\begin{matrix}4 x + y + 3\\x - 4 y - 4\end{matrix}\right] \]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.
- \[ \left[\begin{matrix}6\\\frac{13}{2}\\-34\\- \frac{69}{2}\\\frac{26}{17}\\- \frac{1}{8}\\21\\\frac{217}{8}\\6\end{matrix}\right] \]f at every candidate: (-2, -2), (-2, -3/2), (-2, 3), (-3/2, 3), (-8/17, -19/17), (-1/4, -2), (3, -2), (3, -1/4), (3, 3).✓ Proved
- The largest value is 217/8, the smallest -69/2.
Answer \( \max = \frac{217}{8}\ \text{at}\ (3, - \frac{1}{4});\ \min = - \frac{69}{2}\ \text{at}\ (- \frac{3}{2}, 3) \)
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 2026-10-07qwen3.6:27b-mlx: inconclusive 2026-10-07 — reviewer returned a non-objectqwen3.6:27b-mlx: fail (style) 2026-10-07 — [domain objection, downgraded to style] The solution lists (-2, -3/2) as a candidate point, but y = -3/2 is not a critical point of f on the edge x = -2 (the critical point is y = -1/4). Additionally, the interior critical point (-8/17, -19/17) is outside the domain and should not be included in the comparison.gpt-oss:20b: pass 2026-10-07
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 2026-10-07 with SymPy 1.14.0.