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Horizontal and vertical tangents of parametric curves

Problem 8.351 · medium

Find the points where \( \displaystyle x = t^{3} - 12 t,\ y = t^{3} - 3 t \) 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
\[ \frac{d}{d t} \left(t^{3} - 3 t\right) = 3 t^{2} - 3 \]
dy/dt.✓ Proved
Horizontal tangent: dy/dt = 0 with dx/dt ≠ 0. Vertical: dx/dt = 0 with dy/dt ≠ 0.
Reviewed
\[ \left. 3 t^{2} - 3 \right|_{\substack{ t=-1 }} = 0 \]
t = -1: horizontal, at (11, 2).✓ Proved
\[ \left. 3 t^{2} - 3 \right|_{\substack{ t=1 }} = 0 \]
t = 1: horizontal, at (-11, -2).✓ Proved
\[ \left. 3 t^{2} - 12 \right|_{\substack{ t=-2 }} = 0 \]
t = -2: vertical, at (16, -2).✓ Proved
\[ \left. 3 t^{2} - 12 \right|_{\substack{ t=2 }} = 0 \]
t = 2: vertical, at (-16, 2).✓ Proved

Answer \( \text{horizontal at } (11, 2), (-11, -2);\ \text{vertical at } (16, -2), (-16, 2) \)

✓ Nihil obstat Lines: 6 proved, 1 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	✓ 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
4	✓ Proved	sympy 1.14.0	simplify(lhs - rhs) reduced to 0
5	✓ Proved	sympy 1.14.0	simplify(lhs - rhs) reduced to 0
6	✓ Proved	sympy 1.14.0	simplify(lhs - rhs) reduced to 0
7	✓ 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: pass

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-10-18
gpt-oss:20b: pass 2026-10-18
qwen3.6:27b-mlx: pass 2026-10-16 — The solution correctly identifies the conditions for horizontal and vertical tangents, solves for the critical t-values, and verifies that the other derivative is non-zero at those points. The coordinates are calculated correctly.
gpt-oss:20b: pass 2026-10-16
qwen3.6:27b-mlx: pass 2026-10-14
gpt-oss:20b: pass 2026-10-14

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 2026-10-18 with SymPy 1.14.0.