Home›Calculus 3›Absolute extrema on a closed region›Problem 10.408

Absolute extrema on a closed region

Problem 10.408 · medium

Find the absolute maximum and minimum values of \( \displaystyle f(x, y) = x^{2} - x y - 3 x + 2 y^{2} + 4 y \) on the rectangle \( \displaystyle 0 \le x \le 1 \), \( \displaystyle -2 \le y \le 3 \).

\[ \left[\begin{matrix}\frac{\partial}{\partial x} \left(x^{2} - x y - 3 x + 2 y^{2} + 4 y\right)\\\frac{\partial}{\partial y} \left(x^{2} - x y - 3 x + 2 y^{2} + 4 y\right)\end{matrix}\right] = \left[\begin{matrix}2 x - y - 3\\- x + 4 y + 4\end{matrix}\right] \]
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.
\[ \left[\begin{matrix}0\\-2\\30\\- \frac{1}{4}\\0\\- \frac{25}{8}\\25\end{matrix}\right] \]
f at every candidate: (0, -2), (0, -1), (0, 3), (1/2, -2), (1, -2), (1, -3/4), (1, 3).✓ Proved
The largest value is 30, the smallest -25/8.

Answer \( \max = 30\ \text{at}\ (0, 3);\ \min = - \frac{25}{8}\ \text{at}\ (1, - \frac{3}{4}) \)

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 identify the interior critical point (2, 1/2) where the gradient is zero. Although this point lies outside the specified domain, omitting the step of finding and checking interior critical points is a procedural error in the standard method for finding absolute extrema on a closed region.

Every verdict on record (4)

gpt-oss:20b: pass 2026-10-07
qwen3.6:27b-mlx: fail (style) 2026-10-07 — [domain objection, downgraded to style] The solution fails to identify the interior critical point (2, 1/2) where the gradient is zero. Although this point lies outside the specified domain, omitting the step of finding and checking interior critical points is a procedural error in the standard method for finding absolute extrema on a closed region.
gpt-oss:20b: fail (error) 2026-10-07 — The value of f at (½, –2) was computed incorrectly as 0; the correct value is –¼. This algebraic mistake does not alter the final extrema, but it is a factual error that would mislead a student.
qwen3.6:27b-mlx: pass 2026-10-07 — The solution correctly identifies the method for finding extrema on a closed region (interior critical points and boundary analysis). The listed candidates and their function values are correct, leading to the correct global maximum and minimum.

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.