Home›Calculus 3›Absolute extrema on a closed region›Problem 10.352

Absolute extrema on a closed region

Problem 10.352 · medium

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

\[ \left[\begin{matrix}\frac{\partial}{\partial x} \left(x^{2} + x y + 4 x - 2 y^{2} - y\right)\\\frac{\partial}{\partial y} \left(x^{2} + x y + 4 x - 2 y^{2} - y\right)\end{matrix}\right] = \left[\begin{matrix}2 x + y + 4\\x - 4 y - 1\end{matrix}\right] \]
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.
\[ \left[\begin{matrix}-3\\-15\\12\\\frac{97}{8}\\6\end{matrix}\right] \]
f at every candidate: (-1, 0), (-1, 2), (2, 0), (2, 1/4), (2, 2).✓ Proved
The largest value is 97/8, the smallest -15.

Answer \( \max = \frac{97}{8}\ \text{at}\ (2, \frac{1}{4});\ \min = -15\ \text{at}\ (-1, 2) \)

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 (style) — [domain objection, downgraded to style] The solution fails to find the interior critical point by solving the system 2x + y + 4 = 0 and x - 4y - 1 = 0, which yields (-17/7, -5/7). Although this point is outside the domain, omitting the calculation is a procedural error in the standard method for finding extrema. Additionally, the list of candidates in line 3 is incomplete as it does not explicitly show the evaluation of the interior critical point (even to discard it) or all boundary critical points (e.g., on x=-1 or y=0 edges if they exist within bounds).

Every verdict on record (4)

qwen3.6:27b-mlx: fail (style) 2026-10-05 — [domain objection, downgraded to style] The solution fails to find the interior critical point by solving the system 2x + y + 4 = 0 and x - 4y - 1 = 0, which yields (-17/7, -5/7). Although this point is outside the domain, omitting the calculation is a procedural error in the standard method for finding extrema. Additionally, the list of candidates in line 3 is incomplete as it does not explicitly show the evaluation of the interior critical point (even to discard it) or all boundary critical points (e.g., on x=-1 or y=0 edges if they exist within bounds).
gpt-oss:20b: pass 2026-10-05
qwen3.6:27b-mlx: fail (error) 2026-10-05 — The solution fails to check the interior critical point. Solving ∇f = 0 yields (-1, 1), which lies inside the rectangle and gives f(-1, 1) = -6. While this is not the global min or max, omitting it makes the procedure incomplete and potentially misleading if the interior point were the extremum.
gpt-oss:20b: pass 2026-10-05

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.