Horizontal and vertical tangents of parametric curves
Problem 8.386 · medium
Find the points where \( \displaystyle x = t^{3} - 3 t,\ y = t^{2} - 4 t - 2 \) has a horizontal or a vertical tangent line.
- \[ \frac{d}{d t} \left(t^{3} - 3 t\right) = 3 t^{2} - 3 \]dx/dt.✓ Proved
- \[ \frac{d}{d t} \left(t^{2} - 4 t - 2\right) = 2 t - 4 \]dy/dt.✓ Proved
- Horizontal tangent: dy/dt = 0 with dx/dt ≠ 0. Vertical: dx/dt = 0 with dy/dt ≠ 0.Reviewed
- \[ \left. 2 t - 4 \right|_{\substack{ t=2 }} = 0 \]t = 2: horizontal, at (2, -6).✓ Proved
- \[ \left. 3 t^{2} - 3 \right|_{\substack{ t=-1 }} = 0 \]t = -1: vertical, at (2, 3).✓ Proved
- \[ \left. 3 t^{2} - 3 \right|_{\substack{ t=1 }} = 0 \]t = 1: vertical, at (-2, -5).✓ Proved
Answer \( \text{horizontal at } (2, -6);\ \text{vertical at } (2, 3), (-2, -5) \)
✓ Nihil obstat Lines: 5 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 |
| 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: passqwen3.6:27b-mlx: pass
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-10-18gpt-oss:20b: pass 2026-10-18qwen3.6:27b-mlx: pass 2026-10-16gpt-oss:20b: pass 2026-10-16qwen3.6:27b-mlx: pass 2026-10-15 — The solution correctly identifies the conditions for horizontal and vertical tangents, solves for the parameter values, and computes the corresponding coordinates accurately.gpt-oss:20b: pass 2026-10-15
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.