Horizontal and vertical tangents of parametric curves
Problem 8.244 · medium
Find the points where \( \displaystyle x = t^{3} - 3 t,\ y = 3 t^{2} - 12 \) 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
- \[ \frac{d}{d t} \left(3 t^{2} - 12\right) = 6 t \]dy/dt.✓ Proved
- Horizontal tangent: dy/dt = 0 with dx/dt ≠ 0. Vertical: dx/dt = 0 with dy/dt ≠ 0.
- \[ \left. 6 t \right|_{\substack{ t=0 }} = 0 \]t = 0: horizontal, at (0, -12).✓ Proved
- \[ \left. 3 t^{2} - 3 \right|_{\substack{ t=-1 }} = 0 \]t = -1: vertical, at (2, -9).✓ Proved
- \[ \left. 3 t^{2} - 3 \right|_{\substack{ t=1 }} = 0 \]t = 1: vertical, at (-2, -9).✓ Proved
Answer \( \text{horizontal at } (0, -12);\ \text{vertical at } (2, -9), (-2, -9) \)
Lines: 5 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 |
| 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: inconclusive — reviewer response could not be parsed: {"verdict": "fail", "severity": "error", "notes": "The solution incorrectly identifies t = -1 and t = 1 as points with vertical tangents. At these values, dx/dt = 0 but dy/dt ≠ 0, which indeed indicat
Every verdict on record (6)
qwen3.6:27b-mlx: inconclusive 2026-10-17 — reviewer response could not be parsed: {"verdict": "fail", "severity": "error", "notes": "The solution incorrectly identifies t = -1 and t = 1 as points with vertical tangents. At these values, dx/dt = 0 but dy/dt ≠ 0, which indeed indicatgpt-oss:20b: pass 2026-10-17qwen3.6:27b-mlx: inconclusive 2026-10-15 — reviewer response could not be parsed: {"verdict": "fail", "severity": "error", "notes": "The solution incorrectly identifies vertical tangents at t = ±1. At these points, dy/dt = ±6 ≠ 0, so the tangents are indeed vertical, but the coordigpt-oss:20b: pass 2026-10-15qwen3.6:27b-mlx: fail (error) 2026-10-14 — The solution incorrectly identifies the points for vertical tangents. It solves dx/dt = 0 to get t = ±1, but then incorrectly maps t = -1 to (2, -9) and t = 1 to (-2, -9). In reality, t = 1 gives (x,y) = (-2, -9) and t = -1 gives (x,y) = (2, -9). While the final set of points is correct, the intermediate mapping is swapped, which is a calculation error in the solution steps.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 2026-10-17 with SymPy 1.14.0.