Home›Calculus 3›Absolute extrema on a closed region›Problem 10.354

Absolute extrema on a closed region

Problem 10.354 · medium

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

\[ \left[\begin{matrix}\frac{\partial}{\partial x} \left(- 2 x^{2} + 2 y^{2} + 3 y\right)\\\frac{\partial}{\partial y} \left(- 2 x^{2} + 2 y^{2} + 3 y\right)\end{matrix}\right] = \left[\begin{matrix}- 4 x\\4 y + 3\end{matrix}\right] \]
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.
\[ \left[\begin{matrix}-6\\- \frac{73}{8}\\6\\2\\- \frac{9}{8}\\14\\-16\\- \frac{153}{8}\\-4\end{matrix}\right] \]
f at every candidate: (-2, -2), (-2, -3/4), (-2, 2), (0, -2), (0, -3/4), (0, 2), (3, -2), (3, -3/4), (3, 2).✓ Proved
The largest value is 14, the smallest -153/8.

Answer \( \max = 14\ \text{at}\ (0, 2);\ \min = - \frac{153}{8}\ \text{at}\ (3, - \frac{3}{4}) \)

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 (error) — The solution fails to identify the interior critical point (0, -3/4) in the text, although its value is included in the list. More critically, it omits checking the critical points on the edges x=-2 and x=3 (where y=-3/4 is valid), relying instead on a list of values that includes points not explicitly derived in the text, making the method opaque and incomplete.

Every verdict on record (4)

qwen3.6:27b-mlx: fail (error) 2026-10-05 — The solution fails to identify the interior critical point (0, -3/4) in the text, although its value is included in the list. More critically, it omits checking the critical points on the edges x=-2 and x=3 (where y=-3/4 is valid), relying instead on a list of values that includes points not explicitly derived in the text, making the method opaque and incomplete.
gpt-oss:20b: pass 2026-10-05
qwen3.6:27b-mlx: fail (error) 2026-10-05 — The solution fails to identify the interior critical point (0, -3/4) as a saddle point and incorrectly omits it from the list of candidates in step 3, although its value is coincidentally present. More critically, the logic in step 2 implies checking edges, but step 3 lists points that are not all on the boundary (e.g., (0, -3/4) is interior), making the method description inconsistent with the execution. However, the primary error is that the solution does not explicitly state that the interior critical point is a saddle point and thus not an extremum, nor does it correctly categorize the candidates as 'interior critical points' vs 'boundary points'. The list in step 3 mixes interior and boundary points without clear justification, and the value -153/8 is indeed the global minimum, but the reasoning is flawed because it treats the interior saddle point as just another candidate without explaining why it's included or excluded from the final max/min determination properly. Actually, looking closer, the value -153/8 is at (3, -3/4), which is on the boundary. The point (0, -3/4) is the interior critical point with value -9/8. The solution lists both. The error is subtle: the solution claims to find critical points on edges, but includes an interior point in the list without distinguishing it. This is misleading pedagogy. But is it an error? The final answer is correct. The method described in sentence 2 is 'On each edge... find its critical points... and include corners'. It does NOT mention interior critical points. Yet step 3 includes (0, -3/4) and (0, -2) etc. Wait, (0, -2) is on the boundary. (0, -3/4) is interior. The solution text says 'On each edge...'. It completely ignores the instruction to check interior critical points in the text, but includes them in the calculation. This is a 'misleading' or 'error' in the explanation. A student would learn to just plug in random points. Let's call it an error in the method description.
gpt-oss:20b: pass 2026-10-05

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 2026-10-05 with SymPy 1.14.0.