Home›Calculus 3›Absolute extrema on a closed region›Problem 10.407

Absolute extrema on a closed region

Problem 10.407 · medium

Find the absolute maximum and minimum values of \( \displaystyle f(x, y) = - x^{2} + x y - x + y^{2} - 3 y \) on the rectangle \( \displaystyle 0 \le x \le 1 \), \( \displaystyle 0 \le y \le 2 \).

\[ \left[\begin{matrix}\frac{\partial}{\partial x} \left(- x^{2} + x y - x + y^{2} - 3 y\right)\\\frac{\partial}{\partial y} \left(- x^{2} + x y - x + y^{2} - 3 y\right)\end{matrix}\right] = \left[\begin{matrix}- 2 x + y - 1\\x + 2 y - 3\end{matrix}\right] \]
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.
\[ \left[\begin{matrix}0\\- \frac{9}{4}\\-2\\- \frac{11}{5}\\- \frac{7}{4}\\-2\\-3\\-2\end{matrix}\right] \]
f at every candidate: (0, 0), (0, 3/2), (0, 2), (1/5, 7/5), (1/2, 2), (1, 0), (1, 1), (1, 2).✓ Proved
The largest value is 0, the smallest -3.

Answer \( \max = 0\ \text{at}\ (0, 0);\ \min = -3\ \text{at}\ (1, 1) \)

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 (1/2, 1) because it does not solve the system ∇f = 0, leading to an incomplete list of candidates. Additionally, the stated answer claims the minimum is -3 at (1, 1), but f(1, 1) = -2, whereas the actual minimum is -2.25 at (1/2, 1).

Every verdict on record (4)

gpt-oss:20b: pass 2026-10-07
qwen3.6:27b-mlx: fail (error) 2026-10-07 — The solution fails to identify the interior critical point (1/2, 1) because it does not solve the system ∇f = 0, leading to an incomplete list of candidates. Additionally, the stated answer claims the minimum is -3 at (1, 1), but f(1, 1) = -2, whereas the actual minimum is -2.25 at (1/2, 1).
gpt-oss:20b: pass 2026-10-07
qwen3.6:27b-mlx: fail (error) 2026-10-07 — The solution lists (1, 1) as a candidate point, but (1, 1) is not a critical point on any edge nor a corner; it is an interior point where the gradient is non-zero. The actual interior critical point (1/2, 3/2) is missing from the candidate list, leading to an incorrect minimum value.

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 2026-10-07 with SymPy 1.14.0.