Home›Calculus 3›Absolute extrema on a closed region›Problem 10.305

Absolute extrema on a closed region

Problem 10.305 · medium

Find the absolute maximum and minimum values of \( \displaystyle f(x, y) = x^{2} - x y + 2 x + 2 y^{2} - 4 y \) on the rectangle \( \displaystyle 0 \le x \le 1 \), \( \displaystyle -1 \le y \le 2 \).

\[ \left[\begin{matrix}\frac{\partial}{\partial x} \left(x^{2} - x y + 2 x + 2 y^{2} - 4 y\right)\\\frac{\partial}{\partial y} \left(x^{2} - x y + 2 x + 2 y^{2} - 4 y\right)\end{matrix}\right] = \left[\begin{matrix}2 x - y + 2\\- x + 4 y - 4\end{matrix}\right] \]
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.
\[ \left[\begin{matrix}6\\-2\\0\\10\\- \frac{1}{8}\\1\end{matrix}\right] \]
f at every candidate: (0, -1), (0, 1), (0, 2), (1, -1), (1, 5/4), (1, 2).✓ Proved
The largest value is 10, the smallest -2.

Answer \( \max = 10\ \text{at}\ (1, -1);\ \min = -2\ \text{at}\ (0, 1) \)

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 (0, 1) by solving ∇f = 0, although it correctly lists (0, 1) as a candidate in the final table. This omission makes the derivation of the candidate list logically incomplete and misleading.

Every verdict on record (4)

qwen3.6:27b-mlx: fail (error) 2026-10-04 — The solution fails to identify the interior critical point (0, 1) by solving ∇f = 0, although it correctly lists (0, 1) as a candidate in the final table. This omission makes the derivation of the candidate list logically incomplete and misleading.
gpt-oss:20b: pass 2026-10-04
qwen3.6:27b-mlx: fail (error) 2026-10-04 — The solution fails to find or evaluate the interior critical point (x=0, y=1), which yields the global minimum. Although the final answer happens to be correct, the method described in step 2 implies checking only the boundary, omitting the necessary check for interior critical points.
gpt-oss:20b: pass 2026-10-04

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 2026-10-04 with SymPy 1.14.0.