Home›Calculus 3›Absolute extrema on a closed region›Problem 10.349

Absolute extrema on a closed region

Problem 10.349 · medium

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

\[ \left[\begin{matrix}\frac{\partial}{\partial x} \left(2 x^{2} - x y - 3 x - y^{2} + y\right)\\\frac{\partial}{\partial y} \left(2 x^{2} - x y - 3 x - y^{2} + y\right)\end{matrix}\right] = \left[\begin{matrix}4 x - y - 3\\- x - 2 y + 1\end{matrix}\right] \]
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.
\[ \left[\begin{matrix}10\\\frac{65}{4}\\16\\- \frac{5}{2}\\- \frac{10}{9}\\- \frac{41}{8}\\2\\\frac{9}{4}\\-4\end{matrix}\right] \]
f at every candidate: (-2, -1), (-2, 3/2), (-2, 2), (1/2, -1), (7/9, 1/9), (5/4, 2), (2, -1), (2, -1/2), (2, 2).✓ Proved
The largest value is 65/4, the smallest -41/8.

Answer \( \max = \frac{65}{4}\ \text{at}\ (-2, \frac{3}{2});\ \min = - \frac{41}{8}\ \text{at}\ (\frac{5}{4}, 2) \)

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 (1/2, -1/2) as a candidate for extrema. Although the value at this point is not the global max or min, omitting it violates the standard procedure for finding absolute extrema on a closed region, which requires checking all interior critical points. Furthermore, the list of candidates in step 3 includes (7/9, 1/9), which is not a critical point on any edge nor a corner, suggesting a calculation error or confusion in the boundary analysis.

Every verdict on record (4)

qwen3.6:27b-mlx: fail (error) 2026-10-05 — The solution fails to identify the interior critical point (1/2, -1/2) as a candidate for extrema. Although the value at this point is not the global max or min, omitting it violates the standard procedure for finding absolute extrema on a closed region, which requires checking all interior critical points. Furthermore, the list of candidates in step 3 includes (7/9, 1/9), which is not a critical point on any edge nor a corner, suggesting a calculation error or confusion in the boundary analysis.
gpt-oss:20b: pass 2026-10-05
qwen3.6:27b-mlx: inconclusive 2026-10-05 — reviewer returned a non-object
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.