Home›Calculus 3›Absolute extrema on a closed region›Problem 10.454

Absolute extrema on a closed region

Problem 10.454 · medium

Find the absolute maximum and minimum values of \( \displaystyle f(x, y) = x^{2} - x y + 4 x + y^{2} - 4 y \) on the rectangle \( \displaystyle 0 \le x \le 2 \), \( \displaystyle 0 \le y \le 3 \).

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

Answer \( \max = 12\ \text{at}\ (2, 0);\ \min = -4\ \text{at}\ (0, 2) \)

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: pass
qwen3.6:27b-mlx: fail (error) — The solution fails to identify the interior critical point (x=0, y=2) by solving the system derived in step 1. Although the point (0,2) is included in the final list of candidates, the method described in step 2 ('On each edge...') does not account for interior points, making the logical derivation of the candidate set incomplete and misleading.

Every verdict on record (4)

gpt-oss:20b: pass 2026-10-09
qwen3.6:27b-mlx: fail (error) 2026-10-09 — The solution fails to identify the interior critical point (x=0, y=2) by solving the system derived in step 1. Although the point (0,2) is included in the final list of candidates, the method described in step 2 ('On each edge...') does not account for interior points, making the logical derivation of the candidate set incomplete and misleading.
gpt-oss:20b: pass 2026-10-09
qwen3.6:27b-mlx: fail (error) 2026-10-09 — The solution fails to identify the interior critical point (x=0, y=2) by solving ∇f=0, instead only listing boundary points and corners. While the final numerical answer happens to be correct because the global min occurs on the boundary, the method described is incomplete and would fail for functions where the extremum is in the interior.

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.