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Local extrema and saddle points

Problem 10.220 · easy Video

Find and classify the critical points of \( \displaystyle f(x, y) = x^{2} - 6 x + y^{2} + 2 y + 13 \).

\[ \left[\begin{matrix}\frac{\partial}{\partial x} \left(x^{2} - 6 x + y^{2} + 2 y + 13\right)\\\frac{\partial}{\partial y} \left(x^{2} - 6 x + y^{2} + 2 y + 13\right)\end{matrix}\right] = \left[\begin{matrix}2 x - 6\\2 y + 2\end{matrix}\right] \]
The partial derivatives.✓ Proved
\[ \left[\begin{matrix}0\\0\end{matrix}\right] \]
Both vanish at (3, -1), the only solution.Reviewed
\[ 4 \]
D = f_xx f_yy − f_xy².Reviewed
D = 4 > 0 with f_xx > 0: a local minimum.
Reviewed

Answer \( \text{local minimum at } (3, -1) \)

A contour map of f(x, y) = x² - 6x + y² + 2y + 13: each curve joins points where f has the same value. The contours close in around the critical point (3, -1).

✓ Nihil obstat Lines: 1 proved, 3 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. How we verify

The full receipt

Line	Status	Checked by	Detail
1	✓ Proved	sympy 1.14.0	simplify(lhs - rhs) reduced to 0
2	Reviewed	reviewers: gpt-oss:20b, qwen3.6:27b-mlx	both sides are the same expression: the line states a value, and the working that produced it is not on the line
3	Reviewed	reviewers: gpt-oss:20b, qwen3.6:27b-mlx	both sides are the same expression: the line states a value, and the working that produced it is not on the line
4	Reviewed	reviewers: gpt-oss:20b, qwen3.6:27b-mlx	a sentence; read, not computed
answer, a second way	✓ Checked independently	sympy 1.14.0 + mpmath 1.3.0	f on a small circle around the point is compared with its centre value

Reviewers

gpt-oss:20b: pass
qwen3.6:27b-mlx: pass

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-10-03
gpt-oss:20b: pass 2026-10-03
qwen3.6:27b-mlx: pass 2026-09-28 — The solution correctly identifies the critical point and applies the second derivative test with valid hypotheses and conclusion.
gpt-oss:20b: pass 2026-09-28
qwen3.6:27b-mlx: pass 2026-09-28 — The solution correctly identifies the critical point and applies the second derivative test with valid hypotheses and correct conclusion.
gpt-oss:20b: pass 2026-09-28

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/critical_points_2var, checked 2026-10-13 with SymPy 1.14.0.