Home›Calculus 3›Absolute extrema on a closed region›Problem 10.353

Absolute extrema on a closed region

Problem 10.353 · medium

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

\[ \left[\begin{matrix}\frac{\partial}{\partial x} \left(- 2 x^{2} + x y + y^{2} - 2 y\right)\\\frac{\partial}{\partial y} \left(- 2 x^{2} + x y + y^{2} - 2 y\right)\end{matrix}\right] = \left[\begin{matrix}- 4 x + y\\x + 2 y - 2\end{matrix}\right] \]
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.
\[ \left[\begin{matrix}4\\-11\\\frac{17}{2}\\- \frac{8}{9}\\- \frac{7}{8}\\4\\- \frac{9}{4}\\-2\end{matrix}\right] \]
f at every candidate: (-2, -2), (-2, 1), (-1/2, -2), (2/9, 8/9), (1/4, 1), (1, -2), (1, 1/2), (1, 1).✓ Proved
The largest value is 17/2, the smallest -11.

Answer \( \max = \frac{17}{2}\ \text{at}\ (- \frac{1}{2}, -2);\ \min = -11\ \text{at}\ (-2, 1) \)

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 claims to find critical points on the edges, but the list of candidates includes interior points (e.g., (2/9, 8/9) and (1, 1/2) is not an edge critical point for the given function, likely a calculation error or confusion with interior points) and misses the actual edge critical points. Specifically, the interior critical point is (1/3, 2/3), which is not in the list. The point (1, 1/2) is on the edge x=1, but f(1, 1/2) = -2 + 0.5 + 0.25 - 1 = -2.25 = -9/4, which is in the list. However, the point (2/9, 8/9) is an interior point? No, 2/9 is not on the boundary. Wait, let's check the interior critical point: -4x+y=0 => y=4x. x+2(4x)-2=0 => 9x=2 => x=2/9, y=8/9. This is an interior point. The solution lists it as a candidate, which is correct for global extrema, but the sentence says 'On each edge... find its critical points there too'. It fails to explicitly state that interior critical points are also checked, or it conflates them. More importantly, the point (1/4, 1) is listed. Let's check if (1/4, 1) is a critical point on an edge. On y=1, f(x,1) = -2x^2 + x + 1 - 2 = -2x^2 + x - 1. f' = -4x + 1 = 0 => x=1/4. So (1/4, 1) is a valid edge critical point. The list seems to contain: (-2,-2) corner, (-2,1) corner, (-1/2, -2) edge crit? On y=-2, f(x,-2) = -2x^2 -2x + 4 + 4 = -2x^2 -2x + 8. f' = -4x - 2 = 0 => x = -1/2. Yes. (2/9, 8/9) interior crit. (1/4, 1) edge crit. (1, -2) corner. (1, 1/2) edge crit? On x=1, f(1,y) = -2 + y + y^2 - 2y = y^2 - y - 2. f' = 2y - 1 = 0 => y=1/2. Yes. (1,1) corner. The list of candidates is actually correct and complete for the rectangle. The values are also correct. The sentence 'On each edge... include the four corners' is slightly incomplete because it doesn't explicitly mention checking interior critical points, but the equation line includes them. Is this an error? The sentence describes the method. If the method described doesn't match the execution (which includes interior points), it's misleading or an error in the description. However, usually 'find absolute max/min' implies checking interior and boundary. The sentence only mentions edges and corners. This is a significant omission in the stated method, leading a student to believe interior points are not checked. Thus, it is misleading/error in the setup description.

Every verdict on record (4)

qwen3.6:27b-mlx: fail (error) 2026-10-05 — The solution claims to find critical points on the edges, but the list of candidates includes interior points (e.g., (2/9, 8/9) and (1, 1/2) is not an edge critical point for the given function, likely a calculation error or confusion with interior points) and misses the actual edge critical points. Specifically, the interior critical point is (1/3, 2/3), which is not in the list. The point (1, 1/2) is on the edge x=1, but f(1, 1/2) = -2 + 0.5 + 0.25 - 1 = -2.25 = -9/4, which is in the list. However, the point (2/9, 8/9) is an interior point? No, 2/9 is not on the boundary. Wait, let's check the interior critical point: -4x+y=0 => y=4x. x+2(4x)-2=0 => 9x=2 => x=2/9, y=8/9. This is an interior point. The solution lists it as a candidate, which is correct for global extrema, but the sentence says 'On each edge... find its critical points there too'. It fails to explicitly state that interior critical points are also checked, or it conflates them. More importantly, the point (1/4, 1) is listed. Let's check if (1/4, 1) is a critical point on an edge. On y=1, f(x,1) = -2x^2 + x + 1 - 2 = -2x^2 + x - 1. f' = -4x + 1 = 0 => x=1/4. So (1/4, 1) is a valid edge critical point. The list seems to contain: (-2,-2) corner, (-2,1) corner, (-1/2, -2) edge crit? On y=-2, f(x,-2) = -2x^2 -2x + 4 + 4 = -2x^2 -2x + 8. f' = -4x - 2 = 0 => x = -1/2. Yes. (2/9, 8/9) interior crit. (1/4, 1) edge crit. (1, -2) corner. (1, 1/2) edge crit? On x=1, f(1,y) = -2 + y + y^2 - 2y = y^2 - y - 2. f' = 2y - 1 = 0 => y=1/2. Yes. (1,1) corner. The list of candidates is actually correct and complete for the rectangle. The values are also correct. The sentence 'On each edge... include the four corners' is slightly incomplete because it doesn't explicitly mention checking interior critical points, but the equation line includes them. Is this an error? The sentence describes the method. If the method described doesn't match the execution (which includes interior points), it's misleading or an error in the description. However, usually 'find absolute max/min' implies checking interior and boundary. The sentence only mentions edges and corners. This is a significant omission in the stated method, leading a student to believe interior points are not checked. Thus, it is misleading/error in the setup description.
gpt-oss:20b: pass 2026-10-05
qwen3.6:27b-mlx: pass 2026-10-05 — The solution correctly identifies the method for finding extrema on a closed region (interior critical points, boundary critical points, and corners). The listed candidates and their function values are correct, leading to the correct global maximum and minimum.
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.