Home›Calculus 3›Absolute extrema on a closed region›Problem 10.502

Absolute extrema on a closed region

Problem 10.502 · medium

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

\[ \left[\begin{matrix}\frac{\partial}{\partial x} \left(2 x^{2} - x y + 2 x - 2 y^{2}\right)\\\frac{\partial}{\partial y} \left(2 x^{2} - x y + 2 x - 2 y^{2}\right)\end{matrix}\right] = \left[\begin{matrix}4 x - y + 2\\- x - 4 y\end{matrix}\right] \]
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.
\[ \left[\begin{matrix}-10\\1 \cdot \frac{1}{8}\\-15\\- \frac{8}{17}\\- \frac{145}{8}\\-2\\\frac{33}{8}\\-17\end{matrix}\right] = \left[\begin{matrix}-10\\\frac{1}{8}\\-15\\- \frac{8}{17}\\- \frac{145}{8}\\-2\\\frac{33}{8}\\-17\end{matrix}\right] \]
f at every candidate: (-1, -2), (-1, 1/4), (-1, 3), (-8/17, 2/17), (1/4, 3), (1, -2), (1, -1/4), (1, 3).✓ Proved
The largest value is 33/8, the smallest -145/8.

Answer \( \max = \frac{33}{8}\ \text{at}\ (1, - \frac{1}{4});\ \min = - \frac{145}{8}\ \text{at}\ (\frac{1}{4}, 3) \)

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 (misleading) — The solution lists only boundary candidates but omits the interior critical point (−8/17, 2/17). While that point is a saddle and not an extremum, the omission could mislead a student into thinking the interior has no critical points at all. The explanation should explicitly note that the interior critical point is a saddle and therefore not a candidate for the absolute extrema.
qwen3.6:27b-mlx: fail (error) — The solution fails to evaluate the function at the interior critical point (-1/3, -1/6), which is a required step for finding absolute extrema on a closed region. Although the interior point's value (-11/9) is not the global max or min, omitting it makes the procedure incomplete and teaches that interior critical points can be ignored.

Every verdict on record (4)

qwen3.6:27b-mlx: fail (error) 2026-10-11 — The solution fails to evaluate the function at the interior critical point (-1/3, -1/6), which is a required step for finding absolute extrema on a closed region. Although the interior point's value (-11/9) is not the global max or min, omitting it makes the procedure incomplete and teaches that interior critical points can be ignored.
gpt-oss:20b: fail (misleading) 2026-10-11 — The solution lists only boundary candidates but omits the interior critical point (−8/17, 2/17). While that point is a saddle and not an extremum, the omission could mislead a student into thinking the interior has no critical points at all. The explanation should explicitly note that the interior critical point is a saddle and therefore not a candidate for the absolute extrema.
gpt-oss:20b: pass 2026-10-11
qwen3.6:27b-mlx: pass 2026-10-11 — The solution correctly identifies the method for finding extrema on a closed region (interior critical points and boundary analysis). The listed candidates and their corresponding function values are correct, leading to the correct absolute 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 2026-10-11 with SymPy 1.14.0.