Home›Calculus 3›Absolute extrema on a closed region›Problem 10.302

Absolute extrema on a closed region

Problem 10.302 · medium

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

\[ \left[\begin{matrix}\frac{\partial}{\partial x} \left(x^{2} - x y + 2 x + 2 y^{2}\right)\\\frac{\partial}{\partial y} \left(x^{2} - x y + 2 x + 2 y^{2}\right)\end{matrix}\right] = \left[\begin{matrix}2 x - y + 2\\- x + 4 y\end{matrix}\right] \]
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.
\[ \left[\begin{matrix}4\\- \frac{1}{2}\\4\\- \frac{8}{7}\\\frac{7}{4}\\29\\\frac{111}{8}\\14\end{matrix}\right] \]
f at every candidate: (-2, -2), (-2, -1/2), (-2, 1), (-8/7, -2/7), (-1/2, 1), (3, -2), (3, 3/4), (3, 1).✓ Proved
The largest value is 29, the smallest -8/7.

Answer \( \max = 29\ \text{at}\ (3, -2);\ \min = - \frac{8}{7}\ \text{at}\ (- \frac{8}{7}, - \frac{2}{7}) \)

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 (error) — The value of f at the corner (-2,-2) is 12, not 4 as listed. The table of candidate values is incorrect, so the conclusion about the minimum and maximum is unreliable.
qwen3.6:27b-mlx: pass — The solution correctly identifies the interior critical point and evaluates the function at all necessary candidates (interior critical point, boundary critical points, and corners). The final comparison of values is correct.

Every verdict on record (4)

qwen3.6:27b-mlx: pass 2026-10-04 — The solution correctly identifies the interior critical point and evaluates the function at all necessary candidates (interior critical point, boundary critical points, and corners). The final comparison of values is correct.
gpt-oss:20b: fail (error) 2026-10-04 — The value of f at the corner (-2,-2) is 12, not 4 as listed. The table of candidate values is incorrect, so the conclusion about the minimum and maximum is unreliable.
qwen3.6:27b-mlx: fail (error) 2026-10-04 — The solution lists candidate points for the boundary analysis that are not all valid or complete. Specifically, it includes (-1/2, 1) which is not a critical point on any edge (the critical point on y=1 is at x=-1/2, but f(-1/2, 1) = 7/4 is listed, however the point (-2, -1/2) is listed as a candidate; let's check the edges. Edge x=-2: f(-2,y) = 4+2y-4+2y^2 = 2y^2+2y. Crit pt y=-1/2. f(-2,-1/2) = 2(1/4)-1 = -1/2. This is in the list. Edge x=3: f(3,y) = 9-3y+6+2y^2 = 2y^2-3y+15. Crit pt y=3/4. f(3,3/4) = 2(9/16)-9/4+15 = 9/8 - 18/8 + 120/8 = 111/8 = 13.875. This is in the list. Edge y=-2: f(x,-2) = x^2+2x+2x+8 = x^2+4x+8. Crit pt x=-2. f(-2,-2) = 4. This is in the list. Edge y=1: f(x,1) = x^2-x+2x+2 = x^2+x+2. Crit pt x=-1/2. f(-1/2,1) = 1/4-1/2+2 = 7/4. This is in the list. Corners: (-2,-2)=4, (-2,1)=4, (3,-2)=29, (3,1)=14. Interior: (-8/7, -2/7) = -8/7. The list of values is [4, -1/2, 4, -8/7, 7/4, 29, 111/8, 14]. The max is 29 and min is -8/7. The solution is actually correct in its conclusion. Why did I think it was wrong? Let me re-read carefully. The points listed are: (-2, -2), (-2, -1/2), (-2, 1), (-8/7, -2/7), (-1/2, 1), (3, -2), (3, 3/4), (3, 1). These correspond to: Corner, Edge Crit, Corner, Interior Crit, Edge Crit, Corner, Edge Crit, Corner. All candidates are accounted for. The values match. The conclusion is correct. Wait, is there any subtle error? The problem asks for absolute max/min. The method is standard. The arithmetic in the equation line is marked 'unchecked' but the prompt says 'do not re-check the algebra of a line marked proved'. However, line 3 is marked 'unchecked'. The prompt says 'Each line... is either an EQUATION, which a computer algebra system has already checked... or a SENTENCE'. Line 3 is an equation. It is marked 'unchecked' in the text provided by the user? No, the user text says '[equation, unchecked]'. This usually implies the CAS did NOT check it, or the status is unknown. But the instructions say 'do not re-check the algebra of a line marked proved'. It does not explicitly forbid checking an 'unchecked' line. However, usually in these tasks, if it's an equation line, we assume the algebra is correct unless it's obviously wrong. Let's look at the sentences. Sentence 2: 'On each edge f is a function of one variable: find its critical points there too, and include the four corners.' This is a correct statement of the method. Sentence 4: 'The largest value is 29, the smallest -8/7.' This follows from the list. Is there a missing candidate? No. Are the bounds correct? -2 <= x <= 3, -2 <= y <= 1. Yes. Is the interior point inside? x = -8/7 approx -1.14, y = -2/7 approx -0.28. Inside. Is the max at a corner? (3, -2) is a corner. Is the min at the interior? Yes. The solution seems sound. Why would it fail? Maybe the prompt implies I should check the 'unchecked' equation? If I check the values: f(-2,-2)=4. f(-2,-1/2)=-0.5. f(-2,1)=4. f(-8/7,-2/7)=-8/7. f(-1/2,1)=1.75. f(3,-2)=29. f(3,3/4)=13.875. f(3,1)=14. The list in line 3 is [4, -1/2, 4, -8/7, 7/4, 29, 111/8, 14]. 111/8 = 13.875. 7/4 = 1.75. The values are correct. The points are correct. The conclusion is correct. I will mark it pass.
gpt-oss:20b: pass 2026-10-04

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 2026-10-04 with SymPy 1.14.0.