Absolute extrema on a closed region
Problem 10.456 · medium
Find the absolute maximum and minimum values of \( \displaystyle f(x, y) = 2 x^{2} + x y + 2 x - y^{2} \) on the rectangle \( \displaystyle -2 \le x \le 2 \), \( \displaystyle -2 \le y \le 1 \).
- \[ \left[\begin{matrix}\frac{\partial}{\partial x} \left(2 x^{2} + x y + 2 x - y^{2}\right)\\\frac{\partial}{\partial y} \left(2 x^{2} + x y + 2 x - y^{2}\right)\end{matrix}\right] = \left[\begin{matrix}4 x + y + 2\\x - 2 y\end{matrix}\right] \]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.
- \[ \left[\begin{matrix}4\\5\\1\\- \frac{17}{8}\\- \frac{4}{9}\\-4\\4\\13\end{matrix}\right] \]f at every candidate: (-2, -2), (-2, -1), (-2, 1), (-3/4, 1), (-4/9, -2/9), (0, -2), (2, -2), (2, 1).✓ Proved
- The largest value is 13, the smallest -4.
Answer \( \max = 13\ \text{at}\ (2, 1);\ \min = -4\ \text{at}\ (0, -2) \)
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)
gpt-oss:20b: pass 2026-10-09qwen3.6:27b-mlx: inconclusive 2026-10-09 — reviewer returned a non-objectgpt-oss:20b: pass 2026-10-09qwen3.6:27b-mlx: fail (style) 2026-10-09 — [domain objection, downgraded to style] The solution lists (-3/4, 1) as a candidate point, but x = -3/4 is not a critical point of f(x, 1) = 2x^2 + 3x + 2x - 1 = 2x^2 + 5x - 1, which has derivative 4x + 5, zero at x = -5/4. The value -17/8 corresponds to x = -5/4, so the coordinate is mislabeled. Additionally, the interior critical point (-4/9, -2/9) is inside the domain, but the solution fails to explicitly state that boundary critical points were found by restricting f to the edges and differentiating, instead just listing values.
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.