Absolute extrema on a closed region
Problem 10.565 · medium
Find the absolute maximum and minimum values of \( \displaystyle f(x, y) = - x^{2} + x y + 2 x - y^{2} - 4 y \) on the rectangle \( \displaystyle -2 \le x \le 1 \), \( \displaystyle 0 \le y \le 2 \).
- \[ \left[\begin{matrix}\frac{\partial}{\partial x} \left(- x^{2} + x y + 2 x - y^{2} - 4 y\right)\\\frac{\partial}{\partial y} \left(- x^{2} + x y + 2 x - y^{2} - 4 y\right)\end{matrix}\right] = \left[\begin{matrix}- 2 x + y + 2\\x - 2 y - 4\end{matrix}\right] \]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.
- \[ \left[\begin{matrix}-8\\-24\\1\\-9\end{matrix}\right] \]f at every candidate: (-2, 0), (-2, 2), (1, 0), (1, 2).✓ Proved
- The largest value is 1, the smallest -24.
Answer \( \max = 1\ \text{at}\ (1, 0);\ \min = -24\ \text{at}\ (-2, 2) \)
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: fail (style) — [domain objection, downgraded to style] The solution ignores the interior critical point (0,‑2) which lies outside the domain, but more importantly it fails to examine the edges of the rectangle for one‑variable critical points. Only the four corners are evaluated, so the conclusion about the absolute extrema is unsupported.qwen3.6:27b-mlx: fail (style) — [domain objection, downgraded to style] The solution fails to check the interior critical point. Solving ∇f = 0 yields (x, y) = (-4, -6), which is outside the domain, but the solution does not state this or verify that no interior extrema exist, leaving a gap in the logic for finding absolute extrema.
Every verdict on record (4)
gpt-oss:20b: fail (style) 2026-10-13 — [domain objection, downgraded to style] The solution ignores the interior critical point (0,‑2) which lies outside the domain, but more importantly it fails to examine the edges of the rectangle for one‑variable critical points. Only the four corners are evaluated, so the conclusion about the absolute extrema is unsupported.qwen3.6:27b-mlx: fail (style) 2026-10-13 — [domain objection, downgraded to style] The solution fails to check the interior critical point. Solving ∇f = 0 yields (x, y) = (-4, -6), which is outside the domain, but the solution does not state this or verify that no interior extrema exist, leaving a gap in the logic for finding absolute extrema.gpt-oss:20b: fail (style) 2026-10-13 — [domain objection, downgraded to style] The solution ignores the interior critical point (x=0,y=-2) that lies outside the domain, and it fails to examine the edges of the rectangle. Consequently the set of candidate points is incomplete, leading to an incorrect conclusion about the extrema.qwen3.6:27b-mlx: fail (error) 2026-10-13 — The solution fails to find the interior critical point (x=0, y=-2) and incorrectly omits checking critical points on the edges (e.g., at x=1, y=-1.5 is outside, but on x=-2, y=-1 is outside; however, the method description implies checking edges, but the candidate list in step 3 only includes corners, missing potential edge extrema if they existed within bounds, though in this specific case the edge extrema are at the corners or outside. The primary error is the incomplete description of the method: it claims to find critical points on edges but only lists corners in the evaluation step, which is misleading about the procedure, although numerically the corners happened to be the extrema. Wait, let's re-evaluate. The interior critical point is (0, -2), which is outside the region. The edges: x=-2, f(y) = -4 -2y -4y -y^2 = -y^2-6y-4, vertex at y=-3 (outside). x=1, f(y) = -1+y+2-y^2-4y = -y^2-3y+1, vertex at y=-1.5 (outside). y=0, f(x) = -x^2+2x, vertex at x=1 (corner). y=2, f(x) = -x^2+2x+4-4-4 = -x^2+2x-4, vertex at x=1 (corner). So the extrema are indeed at the corners. However, the sentence in step 2 says 'find its critical points there too', but step 3 only lists the corners. This is a 'misleading' or 'error' in the logic presentation because it skips the actual work of checking edge critical points, implying they don't exist or aren't needed without justification. A student might think you only check corners. This is a pedagogical error.
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 2026-10-13 with SymPy 1.14.0.