Home›Calculus 3›Absolute extrema on a closed region›Problem 10.404

Absolute extrema on a closed region

Problem 10.404 · medium

Find the absolute maximum and minimum values of \( \displaystyle f(x, y) = 2 x^{2} + 4 x + y^{2} + 2 y \) on the rectangle \( \displaystyle -1 \le x \le 3 \), \( \displaystyle -1 \le y \le 1 \).

\[ \left[\begin{matrix}\frac{\partial}{\partial x} \left(2 x^{2} + 4 x + y^{2} + 2 y\right)\\\frac{\partial}{\partial y} \left(2 x^{2} + 4 x + y^{2} + 2 y\right)\end{matrix}\right] = \left[\begin{matrix}4 x + 4\\2 y + 2\end{matrix}\right] \]
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.
\[ \left[\begin{matrix}-3\\1\\29\\33\end{matrix}\right] \]
f at every candidate: (-1, -1), (-1, 1), (3, -1), (3, 1).✓ Proved
The largest value is 33, the smallest -3.

Answer \( \max = 33\ \text{at}\ (3, 1);\ \min = -3\ \text{at}\ (-1, -1) \)

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: pass
qwen3.6:27b-mlx: inconclusive — reviewer response could not be parsed: {"verdict": "fail", "severity": "error", "notes": "The solution fails to evaluate the function at the interior critical point (-1, -1) or check for critical points on the edges (e.g., at x=-1, y=-1 is

Every verdict on record (4)

gpt-oss:20b: pass 2026-10-07
qwen3.6:27b-mlx: inconclusive 2026-10-07 — reviewer response could not be parsed: {"verdict": "fail", "severity": "error", "notes": "The solution fails to evaluate the function at the interior critical point (-1, -1) or check for critical points on the edges (e.g., at x=-1, y=-1 is
qwen3.6:27b-mlx: fail (error) 2026-10-07 — The solution fails to evaluate the function at the interior critical point (-1, -1) explicitly as a candidate, and more critically, it omits checking for critical points on the edges of the rectangle (where the derivative with respect to the varying variable is zero but the point is not a corner). For example, on the edge x=3, f(3,y) = 18 + 9 + y^2 + 2y, which has a critical point at y=-1 (a corner), but on the edge y=1, f(x,1) = 2x^2 + 4x + 1 + 2, derivative 4x+4=0 implies x=-1 (a corner). However, the logic presented in step 2 suggests finding critical points on edges, but step 3 only lists corners. While in this specific case the edge critical points coincide with corners, the solution does not demonstrate that it checked for non-corner edge critical points, making the method incomplete and potentially misleading for other problems. Furthermore, the interior critical point (-1,-1) is listed in step 3, but the derivation of it from step 1 is skipped in the narrative flow, and the claim that these are 'every candidate' is false without showing the edge analysis.
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.