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Level curves and domains

Problem 10.755 · easy Video

Find the equation of the level curve of \( \displaystyle f(x, y) = - 4 x^{2} - y^{2} + 6 \) that passes through \( \displaystyle (0, 1) \), and describe it.

A level curve is f(x, y) = c; the point fixes c.
Reviewed
\[ \left. - 4 x^{2} - y^{2} + 6 \right|_{\substack{ x=0\\ y=1 }} = 5 \]
c = f(0, 1).✓ Proved
So the level curve is Eq(-4*x**2 - y**2 + 6, 5), a ellipse.
Reviewed

Answer \( - 4 x^{2} - y^{2} + 6 = 5\ (\text{a ellipse}) \)

A contour map of f(x, y) = -4x² - y² + 6: each curve joins points where f has the same value. The level curve through (0, 1) is highlighted.

Lines: 1 proved, 2 reviewed. 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. How we verify

The full receipt

Line	Status	Checked by	Detail
1	Reviewed	reviewers: gpt-oss:20b, qwen3.6:27b-mlx	a sentence; read, not computed
2	✓ Proved	sympy 1.14.0	simplify(lhs - rhs) reduced to 0
3	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	points found numerically on the claimed curve near P give f the same value

Reviewers

gpt-oss:20b: pass
qwen3.6:27b-mlx: pass — The solution correctly identifies the constant c by evaluating the function at the given point and correctly classifies the resulting equation as an ellipse.

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-10-31 — The solution correctly identifies the constant c by evaluating the function at the given point and correctly classifies the resulting equation as an ellipse.
gpt-oss:20b: pass 2026-10-31
qwen3.6:27b-mlx: fail (error) 2026-10-30 — The equation -4x^2 - y^2 + 6 = 5 simplifies to 4x^2 + y^2 = 1, which is an ellipse. However, the original function f(x, y) = -4x^2 - y^2 + 6 defines a downward-opening elliptic paraboloid. The level curve at z=5 is indeed an ellipse, but the solution fails to simplify the equation to standard form, leaving it in a non-standard implicit form that obscures the geometric properties (axes lengths, orientation). More critically, while 'ellipse' is correct, the phrasing 'a ellipse' is grammatically poor, but the main issue is that the solution does not explicitly derive the standard form 4x^2 + y^2 = 1, which is the expected 'equation of the level curve'. The stated answer '-4x^2 - y^2 + 6 = 5' is technically an equation of the curve, but it is not simplified. Is this an error? The problem asks for 'the equation', implying a simplified form. Leaving it as f(x,y)=c is often acceptable in early calculus, but describing it as 'a ellipse' without simplifying to show it's bounded and closed might be considered incomplete. However, looking closer, the prompt asks to 'Find the equation... and describe it'. The solution provides an equation and a description. The equation is correct. The description is correct. Is there a mathematical error? No. Is it misleading? No. Is it style? 'a ellipse' is bad style. But is it an error? Usually, 'find the equation' implies simplifying. Let's look at the options. If I mark it 'style', it's weak. If I mark it 'error', it's strong. Let's re-read the prompt constraints. 'Check that... the setup models the problem... right formula...'. The formula is right. The conclusion follows. The only issue is simplification. In many contexts, f(x,y)=c is the definition. But standard practice is to simplify. Let's look for a deeper error. f(0,1) = 5. Level curve is -4x^2 - y^2 + 6 = 5 => 4x^2 + y^2 = 1. This is an ellipse. The solution says 'a ellipse'. This is grammatically incorrect. Is that 'style'? Yes. But wait, is there a sign error? -4(0)^2 - (1)^2 + 6 = 5. Correct. Is it an ellipse? Yes. So the math is correct. The only flaw is the lack of simplification and grammar. I will mark it as style because the math is fundamentally correct, just not presented in standard form.
gpt-oss:20b: pass 2026-10-30
qwen3.6:27b-mlx: pass 2026-10-30 — The solution correctly identifies the constant c by evaluating the function at the given point and sets up the equation for the level curve. The classification of the curve as an ellipse is correct.
gpt-oss:20b: pass 2026-10-30

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/level_curves, checked 2026-10-31 with SymPy 1.14.0.