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Horizontal and vertical tangents of parametric curves

Problem 8.578 · medium

Find the points where \( \displaystyle x = t^{3} - 12 t,\ y = t^{2} - 4 t + 1 \) has a horizontal or a vertical tangent line.

\[ \frac{d}{d t} \left(t^{3} - 12 t\right) = 3 t^{2} - 12 \]
dx/dt.✓ Proved
\[ \frac{d}{d t} \left(t^{2} - 4 t + 1\right) = 2 t - 4 \]
dy/dt.✓ Proved
Horizontal tangent: dy/dt = 0 with dx/dt ≠ 0. Vertical: dx/dt = 0 with dy/dt ≠ 0.
\[ \left. 3 t^{2} - 12 \right|_{\substack{ t=-2 }} = 0 \]
t = -2: vertical, at (16, 13).✓ Proved

Answer \( \text{horizontal at } none;\ \text{vertical at } (16, 13) \)

Lines: 3 proved, 1 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. How we verify

The full receipt

Line	Status	Checked by	Detail
1	✓ Proved	sympy 1.14.0	simplify(lhs - rhs) reduced to 0
2	✓ Proved	sympy 1.14.0	simplify(lhs - rhs) reduced to 0
3	Not checked	—	a sentence; read, not computed
4	✓ Proved	sympy 1.14.0	simplify(lhs - rhs) reduced to 0
answer, a second way	✓ Checked independently	sympy 1.14.0 + mpmath 1.3.0	roots of dx/dt and dy/dt found by Newton's method from a dense grid

Reviewers

gpt-oss:20b: pass
qwen3.6:27b-mlx: fail (error) — The solution fails to check for horizontal tangents. It correctly identifies the vertical tangent at t = -2, but ignores the case where dy/dt = 0 (which occurs at t = 2). At t = 2, dx/dt = -12 ≠ 0, so there is a horizontal tangent at (-16, -3), which is missing from the final answer.

Every verdict on record (6)

qwen3.6:27b-mlx: fail (error) 2026-11-13 — The solution fails to check for horizontal tangents. It correctly identifies the vertical tangent at t = -2, but ignores the case where dy/dt = 0 (which occurs at t = 2). At t = 2, dx/dt = -12 ≠ 0, so there is a horizontal tangent at (-16, -3), which is missing from the final answer.
gpt-oss:20b: pass 2026-11-13
qwen3.6:27b-mlx: fail (error) 2026-11-12 — The solution fails to check for horizontal tangents. It correctly identifies the vertical tangent at t=-2, but omits the analysis for dy/dt = 0 (which occurs at t=2). At t=2, dx/dt is non-zero, so there is a horizontal tangent at (-16, -3), which is missing from the final answer.
gpt-oss:20b: pass 2026-11-12
qwen3.6:27b-mlx: fail (error) 2026-11-12 — The solution fails to check for horizontal tangents, incorrectly concluding there are none without solving dy/dt = 0. It also omits the vertical tangent at t=2, only checking t=-2.
gpt-oss:20b: pass 2026-11-12

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/parametric_tangent_hv, checked 2026-11-13 with SymPy 1.14.0.