Home›Calculus 3›Absolute extrema on a closed region›Problem 10.457

Absolute extrema on a closed region

Problem 10.457 · medium

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

\[ \left[\begin{matrix}\frac{\partial}{\partial x} \left(2 x^{2} - x y + 4 x - 2 y^{2} + 2 y\right)\\\frac{\partial}{\partial y} \left(2 x^{2} - x y + 4 x - 2 y^{2} + 2 y\right)\end{matrix}\right] = \left[\begin{matrix}4 x - y + 4\\- x - 4 y + 2\end{matrix}\right] \]
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.
\[ \left[\begin{matrix}-12\\1 \cdot \frac{1}{2}\\-12\\8\\16\\-2\end{matrix}\right] = \left[\begin{matrix}-12\\\frac{1}{2}\\-12\\8\\16\\-2\end{matrix}\right] \]
f at every candidate: (0, -2), (0, 1/2), (0, 3), (2, -2), (2, 0), (2, 3).✓ Proved
The largest value is 16, the smallest -12.

Answer \( \max = 16\ \text{at}\ (2, 0);\ \min = -12\ \text{at}\ (0, -2), (0, 3) \)

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: fail (error) — The solution omits critical points on the edges x=2 (y=1/2) and y=3 (x=3/4), and therefore does not evaluate all necessary candidates. Consequently the reported extrema are not justified.
qwen3.6:27b-mlx: fail (error) — The solution fails to find the interior critical point at (1, 0), which yields f(1, 0) = 6. While this is not the global max or min, omitting it is a procedural error in finding all critical points. More critically, the solution lists (0, 1/2) as a candidate but does not explain how it was derived (it is a critical point on the edge x=0), making the method opaque and potentially misleading about how boundary extrema are found.

Every verdict on record (4)

gpt-oss:20b: fail (error) 2026-10-09 — The solution omits critical points on the edges x=2 (y=1/2) and y=3 (x=3/4), and therefore does not evaluate all necessary candidates. Consequently the reported extrema are not justified.
qwen3.6:27b-mlx: fail (error) 2026-10-09 — The solution fails to find the interior critical point at (1, 0), which yields f(1, 0) = 6. While this is not the global max or min, omitting it is a procedural error in finding all critical points. More critically, the solution lists (0, 1/2) as a candidate but does not explain how it was derived (it is a critical point on the edge x=0), making the method opaque and potentially misleading about how boundary extrema are found.
gpt-oss:20b: pass 2026-10-09
qwen3.6:27b-mlx: fail (error) 2026-10-09 — The solution fails to find the interior critical point (x=2, y=0), which yields the global maximum. It incorrectly identifies (2,0) as a boundary point in the candidate list, omitting the necessary step of solving the system of equations from Step 1.

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.