Horizontal and vertical tangents of parametric curves
Problem 8.243 · medium
Find the points where \( \displaystyle x = t^{3} - 6 t,\ y = 3 t^{2} - 6 \) has a horizontal or a vertical tangent line.
- \[ \frac{d}{d t} \left(t^{3} - 6 t\right) = 3 t^{2} - 6 \]dx/dt.✓ Proved
- \[ \frac{d}{d t} \left(3 t^{2} - 6\right) = 6 t \]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, -6).✓ Proved
- \[ \left. 3 t^{2} - 6 \right|_{\substack{ t=- \sqrt{2} }} = 0 \]t = -sqrt(2): vertical, at (4*sqrt(2), 0).✓ Proved
- \[ \left. 3 t^{2} - 6 \right|_{\substack{ t=\sqrt{2} }} = 0 \]t = sqrt(2): vertical, at (-4*sqrt(2), 0).✓ Proved
Answer \( \text{horizontal at } (0, -6);\ \text{vertical at } (4*sqrt(2), 0), (-4*sqrt(2), 0) \)
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: fail (error) — The solution incorrectly identifies the points for vertical tangents. It solves dy/dt = 0 (which gives horizontal tangents) instead of dx/dt = 0 (which gives vertical tangents). Consequently, the points listed as having vertical tangents actually have horizontal tangents, and the points with actual vertical tangents are omitted.
Every verdict on record (6)
qwen3.6:27b-mlx: fail (error) 2026-10-17 — The solution incorrectly identifies the points for vertical tangents. It solves dy/dt = 0 (which gives horizontal tangents) instead of dx/dt = 0 (which gives vertical tangents). Consequently, the points listed as having vertical tangents actually have horizontal tangents, and the points with actual vertical tangents are omitted.gpt-oss:20b: pass 2026-10-17qwen3.6:27b-mlx: fail (style) 2026-10-15 — [domain objection, downgraded to style] The solution incorrectly identifies the points for vertical tangents. Vertical tangents occur when dx/dt = 0 (at t = ±sqrt(2)), but the solution substitutes these t-values into the equation for dy/dt (or rather, checks dy/dt=0 which is for horizontal tangents) and labels them vertical. The points calculated are actually where the tangent is horizontal (or undefined if both are zero, but here dy/dt is non-zero at t=±sqrt(2), so they are vertical tangents, but the logic in step 5/6 is garbled: it says 't = -sqrt(2): vertical' but the equation shown is Subs(3*t**2 - 6, ...)=0 which is dy/dt=0, the condition for horizontal tangents. Wait, let's re-read carefully. Step 3 says: Horizontal: dy/dt=0, dx/dt!=0. Vertical: dx/dt=0, dy/dt!=0. Step 4: Checks dy/dt=0 at t=0. dx/dt(0) = -6 != 0. So t=0 is horizontal. Correct. Step 5: Says 't = -sqrt(2): vertical'. But the equation shown is `Subs(3*t**2 - 6, t, -sqrt(2)) = 0`. `3*t**2 - 6` is `dx/dt`. So it is checking `dx/dt = 0`. This is the correct condition for vertical tangents. However, the comment says 'vertical, at (4*sqrt(2), 0)'. Let's check the point. x(-sqrt(2)) = (-sqrt(2))^3 - 6(-sqrt(2)) = -2sqrt(2) + 6sqrt(2) = 4sqrt(2). y(-sqrt(2)) = 3(2) - 6 = 0. So the point is correct. The condition checked is correct (dx/dt=0). Is dy/dt != 0? dy/dt(-sqrt(2)) = -6sqrt(2) != 0. So it is a vertical tangent. Step 6: Says 't = sqrt(2): vertical'. Equation shown is `Subs(3*t**2 - 6, t, sqrt(2)) = 0`. This is `dx/dt = 0`. Correct condition. Point: x(sqrt(2)) = 2sqrt(2) - 6sqrt(2) = -4sqrt(2). y(sqrt(2)) = 0. Point is (-4sqrt(2), 0). Correct. So the math is actually correct. Why did I think it was wrong? I misread the variable names in my head. `3*t**2 - 6` is `dx/dt`. The solution checks `dx/dt=0` for vertical tangents. This is correct. Let's double check Step 3. "Horizontal tangent: dy/dt = 0 with dx/dt ≠ 0. Vertical: dx/dt = 0 with dy/dt ≠ 0." This is the standard definition for parametric curves. Step 4: `Subs(6*t, t, 0) = 0`. `6*t` is `dy/dt`. So it checks `dy/dt=0`. Correct for horizontal. Step 5: `Subs(3*t**2 - 6, t, -sqrt(2)) = 0`. `3*t**2 - 6` is `dx/dt`. So it checks `dx/dt=0`. Correct for vertical. Step 6: `Subs(3*t**2 - 6, t, sqrt(2)) = 0`. Checks `dx/dt=0`. Correct for vertical. The solution is correct. The points are correct. The logic is correct.gpt-oss:20b: pass 2026-10-15qwen3.6:27b-mlx: fail (error) 2026-10-14 — The solution incorrectly identifies the points for vertical tangents. It sets dy/dt = 0 (which finds horizontal tangents) instead of dx/dt = 0. Furthermore, the coordinates calculated for t = ±sqrt(2) are incorrect; x(±sqrt(2)) is ±4*sqrt(2), but the signs are swapped in the final answer relative to the t values used.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.