Horizontal and vertical tangents of parametric curves
Problem 8.577 · medium
Find the points where \( \displaystyle x = t^{3} - 12 t,\ y = t^{2} - 4 t + 3 \) 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
- \[ \frac{d}{d t} \left(t^{2} - 4 t + 3\right) = 2 t - 4 \]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, 15).✓ Proved
Answer \( \text{horizontal at } none;\ \text{vertical at } (16, 15) \)
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: passqwen3.6:27b-mlx: fail (error) — The solution fails to check for horizontal tangents (dy/dt = 0). At t=2, dy/dt=0 and dx/dt=-12, so there is a horizontal tangent at (-20, -1). The stated answer 'none' is incorrect.
Every verdict on record (6)
qwen3.6:27b-mlx: fail (error) 2026-11-13 — The solution fails to check for horizontal tangents (dy/dt = 0). At t=2, dy/dt=0 and dx/dt=-12, so there is a horizontal tangent at (-20, -1). The stated answer 'none' is incorrect.gpt-oss:20b: pass 2026-11-13qwen3.6:27b-mlx: fail (error) 2026-11-12 — The solution fails to check for horizontal tangents by solving dy/dt = 0. It also incorrectly concludes 'none' for horizontal tangents without showing the work, whereas dy/dt = 0 at t=2, where dx/dt != 0, so there is a horizontal tangent at (-16, -5).gpt-oss:20b: pass 2026-11-12qwen3.6:27b-mlx: fail (error) 2026-11-12 — The solution fails to check for horizontal tangents (dy/dt = 0), incorrectly concluding there are none. At t=2, dy/dt=0 and dx/dt=-12≠0, so there is a horizontal tangent at (-20, -1).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.