Home›Calculus 3›Absolute extrema on a closed region›Problem 10.499

Absolute extrema on a closed region

Problem 10.499 · medium

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

\[ \left[\begin{matrix}\frac{\partial}{\partial x} \left(2 x^{2} + 3 x - 2 y^{2} + 3 y\right)\\\frac{\partial}{\partial y} \left(2 x^{2} + 3 x - 2 y^{2} + 3 y\right)\end{matrix}\right] = \left[\begin{matrix}4 x + 3\\3 - 4 y\end{matrix}\right] \]
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.
\[ \left[\begin{matrix}-6\\1 \cdot \frac{1}{8}\\-10\\- \frac{49}{8}\\0\\- \frac{81}{8}\\9\\\frac{121}{8}\\5\end{matrix}\right] = \left[\begin{matrix}-6\\\frac{1}{8}\\-10\\- \frac{49}{8}\\0\\- \frac{81}{8}\\9\\\frac{121}{8}\\5\end{matrix}\right] \]
f at every candidate: (-1, -1), (-1, 3/4), (-1, 3), (-3/4, -1), (-3/4, 3/4), (-3/4, 3), (2, -1), (2, 3/4), (2, 3).✓ Proved
The largest value is 121/8, the smallest -81/8.

Answer \( \max = \frac{121}{8}\ \text{at}\ (2, \frac{3}{4});\ \min = - \frac{81}{8}\ \text{at}\ (- \frac{3}{4}, 3) \)

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: pass
qwen3.6:27b-mlx: fail (error) — The solution fails to identify the interior critical point (x=-3/4, y=3/4) as a candidate for extrema, listing it in the evaluation table but omitting it from the logical setup in step 2. While the final numerical answer happens to be correct because the interior point is a saddle point, the method described is incomplete and would fail for functions where the extremum occurs at an interior critical point.

Every verdict on record (4)

qwen3.6:27b-mlx: fail (error) 2026-10-11 — The solution fails to identify the interior critical point (x=-3/4, y=3/4) as a candidate for extrema, listing it in the evaluation table but omitting it from the logical setup in step 2. While the final numerical answer happens to be correct because the interior point is a saddle point, the method described is incomplete and would fail for functions where the extremum occurs at an interior critical point.
gpt-oss:20b: pass 2026-10-11
gpt-oss:20b: pass 2026-10-11
qwen3.6:27b-mlx: fail (error) 2026-10-11 — The solution fails to identify the interior critical point (x=-3/4, y=3/4) as a candidate for extrema. While the point is included in the list of values in step 3, the text in step 2 explicitly states to find critical points on the edges, omitting the interior critical point which is required by the Extreme Value Theorem for finding absolute extrema on a closed region. This omission in the methodology is a significant error in the explanation.

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.