Home›Calculus 3›Absolute extrema on a closed region›Problem 10.406

Absolute extrema on a closed region

Problem 10.406 · medium

Find the absolute maximum and minimum values of \( \displaystyle f(x, y) = x^{2} + x y - x - y^{2} + 2 y \) on the rectangle \( \displaystyle 0 \le x \le 3 \), \( \displaystyle -1 \le y \le 2 \).

\[ \left[\begin{matrix}\frac{\partial}{\partial x} \left(x^{2} + x y - x - y^{2} + 2 y\right)\\\frac{\partial}{\partial y} \left(x^{2} + x y - x - y^{2} + 2 y\right)\end{matrix}\right] = \left[\begin{matrix}2 x + y - 1\\x - 2 y + 2\end{matrix}\right] \]
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.
\[ \left[\begin{matrix}-3\\1\\0\\-4\\0\\12\end{matrix}\right] \]
f at every candidate: (0, -1), (0, 1), (0, 2), (1, -1), (3, -1), (3, 2).✓ Proved
The largest value is 12, the smallest -4.

Answer \( \max = 12\ \text{at}\ (3, 2);\ \min = -4\ \text{at}\ (1, -1) \)

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: fail (error) — The solution fails to identify the interior critical point (1, 1) and evaluate the function there. Although f(1,1)=2 is not the global extremum, omitting the interior check is a procedural error in finding absolute extrema on a closed region.

Every verdict on record (4)

gpt-oss:20b: pass 2026-10-07
qwen3.6:27b-mlx: fail (error) 2026-10-07 — The solution fails to identify the interior critical point (1, 1) and evaluate the function there. Although f(1,1)=2 is not the global extremum, omitting the interior check is a procedural error in finding absolute extrema on a closed region.
qwen3.6:27b-mlx: fail (error) 2026-10-07 — The solution fails to identify the interior critical point (1, 1), which yields a local maximum of 2. While this is not the global maximum, omitting the standard step of finding and evaluating interior critical points makes the method incomplete and misleading for a student learning the procedure.
gpt-oss:20b: fail (error) 2026-10-07 — The solution omits the interior critical point (0,1) and incorrectly evaluates the minimum. The true minimum occurs at (0.5,‑1) with f=‑3.25, not at (1,‑1) with f=‑4. The edge analysis also misses the point (0.5,‑1).

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.