Home›Calculus 3›Absolute extrema on a closed region›Problem 10.356

Absolute extrema on a closed region

Problem 10.356 · medium

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

\[ \left[\begin{matrix}\frac{\partial}{\partial x} \left(- x^{2} - 2 x + y^{2} - 4 y\right)\\\frac{\partial}{\partial y} \left(- x^{2} - 2 x + y^{2} - 4 y\right)\end{matrix}\right] = \left[\begin{matrix}- 2 x - 2\\2 y - 4\end{matrix}\right] \]
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.
\[ \left[\begin{matrix}5\\-4\\6\\-3\\2\\-7\end{matrix}\right] \]
f at every candidate: (-2, -1), (-2, 2), (-1, -1), (-1, 2), (1, -1), (1, 2).✓ Proved
The largest value is 6, the smallest -7.

Answer \( \max = 6\ \text{at}\ (-1, -1);\ \min = -7\ \text{at}\ (1, 2) \)

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 at (-1, 2), which yields a value of -5. Although this value is not the global maximum or minimum, omitting it from the list of candidates is a procedural error in finding extrema on a closed region, as the method requires checking all critical points (interior and boundary) to guarantee the result.

Every verdict on record (4)

qwen3.6:27b-mlx: fail (error) 2026-10-05 — The solution fails to identify the interior critical point at (-1, 2), which yields a value of -5. Although this value is not the global maximum or minimum, omitting it from the list of candidates is a procedural error in finding extrema on a closed region, as the method requires checking all critical points (interior and boundary) to guarantee the result.
gpt-oss:20b: pass 2026-10-05
qwen3.6:27b-mlx: fail (error) 2026-10-05 — The solution fails to find or evaluate the interior critical point (-1, 2), which yields a value of 6. While the final maximum value is correct, the stated location (-1, -1) is incorrect (f(-1,-1) = 5), and the method described ignores the interior critical point entirely.
gpt-oss:20b: fail (error) 2026-10-05 — The solution omits checking the interior critical point (x=–1,y=2). While the final extrema are correct, the omission means the reasoning is incomplete and could mislead a student into thinking only boundary points need be examined.

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 2026-10-05 with SymPy 1.14.0.