Home›Calculus 3›Absolute extrema on a closed region›Problem 10.566

Absolute extrema on a closed region

Problem 10.566 · medium

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

\[ \left[\begin{matrix}\frac{\partial}{\partial x} \left(- x^{2} - x y + 4 x + 2 y^{2} + y\right)\\\frac{\partial}{\partial y} \left(- x^{2} - x y + 4 x + 2 y^{2} + y\right)\end{matrix}\right] = \left[\begin{matrix}- 2 x - y + 4\\- x + 4 y + 1\end{matrix}\right] \]
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.
\[ \left[\begin{matrix}-12\\-7\\\frac{21}{4}\\\frac{35}{9}\\4\\\frac{31}{8}\\5\end{matrix}\right] \]
f at every candidate: (-2, 0), (-2, 1), (3/2, 1), (17/9, 2/9), (2, 0), (2, 1/4), (2, 1).✓ Proved
The largest value is 21/4, the smallest -12.

Answer \( \max = \frac{21}{4}\ \text{at}\ (\frac{3}{2}, 1);\ \min = -12\ \text{at}\ (-2, 0) \)

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 (17/9, 2/9) as a candidate in the text, listing only boundary points and corners in step 2, although its value is included in step 3. This implies the interior was ignored in the methodology described, which is incorrect for finding absolute extrema.

Every verdict on record (4)

gpt-oss:20b: pass 2026-10-13
qwen3.6:27b-mlx: fail (error) 2026-10-13 — The solution fails to identify the interior critical point (17/9, 2/9) as a candidate in the text, listing only boundary points and corners in step 2, although its value is included in step 3. This implies the interior was ignored in the methodology described, which is incorrect for finding absolute extrema.
gpt-oss:20b: pass 2026-10-13
qwen3.6:27b-mlx: pass 2026-10-13 — The solution correctly identifies the method (checking interior critical points and boundary extrema) and the final values are correct. Although the list of candidates in line 3 includes an interior point (17/9, 2/9) which is not a critical point of the full function, it does not affect the final result as the true extrema are found elsewhere.

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 2026-10-13 with SymPy 1.14.0.