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Horizontal and vertical tangents of parametric curves

Problem 8.238 · medium

Find the points where \( \displaystyle x = t^{3} - 9 t,\ y = 3 t^{2} - 9 \) has a horizontal or a vertical tangent line.
  1. \[ \frac{d}{d t} \left(t^{3} - 9 t\right) = 3 t^{2} - 9 \]
    dx/dt.✓ Proved
  2. \[ \frac{d}{d t} \left(3 t^{2} - 9\right) = 6 t \]
    dy/dt.✓ Proved
  3. Horizontal tangent: dy/dt = 0 with dx/dt ≠ 0. Vertical: dx/dt = 0 with dy/dt ≠ 0.
  4. \[ \left. 6 t \right|_{\substack{ t=0 }} = 0 \]
    t = 0: horizontal, at (0, -9).✓ Proved
  5. \[ \left. 3 t^{2} - 9 \right|_{\substack{ t=- \sqrt{3} }} = 0 \]
    t = -sqrt(3): vertical, at (6*sqrt(3), 0).✓ Proved
  6. \[ \left. 3 t^{2} - 9 \right|_{\substack{ t=\sqrt{3} }} = 0 \]
    t = sqrt(3): vertical, at (-6*sqrt(3), 0).✓ Proved
Answer \( \text{horizontal at } (0, -9);\ \text{vertical at } (6*sqrt(3), 0), (-6*sqrt(3), 0) \)

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
LineStatusChecked byDetail
1✓ Provedsympy 1.14.0simplify(lhs - rhs) reduced to 0
2✓ Provedsympy 1.14.0simplify(lhs - rhs) reduced to 0
3Not checked—a sentence; read, not computed
4✓ Provedsympy 1.14.0simplify(lhs - rhs) reduced to 0
5✓ Provedsympy 1.14.0simplify(lhs - rhs) reduced to 0
6✓ Provedsympy 1.14.0simplify(lhs - rhs) reduced to 0
answer, a second way✓ Checked independentlysympy 1.14.0 + mpmath 1.3.0roots of dx/dt and dy/dt found by Newton's method from a dense grid

Reviewers

  • gpt-oss:20b: pass
  • qwen3.6:27b-mlx: fail (error) — 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(3) are incorrect; substituting these t-values into x(t) yields ±6*sqrt(3), but the solution swaps the signs or values in the final answer relative to the t-values used, and fundamentally misapplies the condition for vertical tangents.
Every verdict on record (6)
  • qwen3.6:27b-mlx: fail (error) 2026-10-17 — 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(3) are incorrect; substituting these t-values into x(t) yields ±6*sqrt(3), but the solution swaps the signs or values in the final answer relative to the t-values used, and fundamentally misapplies the condition for vertical tangents.
  • gpt-oss:20b: pass 2026-10-17
  • qwen3.6:27b-mlx: fail (error) 2026-10-15 — The solution incorrectly identifies the points for vertical tangents. It sets dx/dt = 0 (which occurs at t = ±√3) but then substitutes these t-values into y(t) to find the y-coordinate, while the stated points use x-coordinates derived from t = ±3 (where dy/dt = 0). Specifically, at t = ±√3, x = ±6√3 and y = 0, so the vertical tangents are at (±6√3, 0). However, the solution's logic in steps 5 and 6 is garbled: it claims t = ±√3 are vertical (correct condition dx/dt=0), but the comment says 'vertical at (6*sqrt(3), 0)' for t=-sqrt(3) which is actually x = (-sqrt(3))^3 - 9(-sqrt(3)) = -3sqrt(3) + 9sqrt(3) = 6sqrt(3). Wait, let's re-calculate. x(t) = t^3 - 9t. At t=sqrt(3), x = 3sqrt(3) - 9sqrt(3) = -6sqrt(3). At t=-sqrt(3), x = -3sqrt(3) + 9sqrt(3) = 6sqrt(3). So the points (6sqrt(3), 0) and (-6sqrt(3), 0) are indeed the correct locations for vertical tangents. The error is in the reasoning text: Step 5 says 't = -sqrt(3): vertical... at (6*sqrt(3), 0)'. This is numerically correct. Step 6 says 't = sqrt(3): vertical... at (-6*sqrt(3), 0)'. This is also numerically correct. However, the sentence in Step 3 defines vertical as dx/dt=0. Steps 5 and 6 check dx/dt=0 (by substituting into the derivative expression 3t^2-9? No, they substitute into 3t^2-9 which is dx/dt? No, 3t^2-9 is dx/dt. The equation proved is Subs(3*t**2 - 9, t, ...) = 0. This confirms dx/dt=0. So the logic is actually correct. Let me re-read carefully. Step 1: dx/dt = 3t^2-9. Step 2: dy/dt = 6t. Step 3: Horizontal dy/dt=0, Vertical dx/dt=0. Step 4: Checks dy/dt=0 at t=0. Correct. Step 5: Checks dx/dt=0 at t=-sqrt(3). Correct. Point is (6sqrt(3), 0). Correct. Step 6: Checks dx/dt=0 at t=sqrt(3). Correct. Point is (-6sqrt(3), 0). Correct. The solution is actually correct. Why did I think it was wrong? Ah, I misread the substitution. The equation proved is `Subs(3*t**2 - 9, t, -sqrt(3)) = 0`. This is checking dx/dt. The comment says 'vertical'. This is consistent. The points are correct. The verdict should be pass.
  • gpt-oss:20b: pass 2026-10-15
  • qwen3.6:27b-mlx: fail (error) 2026-10-14 — The solution incorrectly identifies the points for vertical tangents. It uses t = ±sqrt(3) (where dx/dt = 0) but fails to verify that dy/dt ≠ 0 at these points (dy/dt = ±6sqrt(3) ≠ 0, so they are indeed vertical tangents). However, the coordinates calculated in lines 5 and 6 are swapped or incorrect relative to the parameter t. For t = -sqrt(3), x = (-sqrt(3))^3 - 9(-sqrt(3)) = -3sqrt(3) + 9sqrt(3) = 6sqrt(3), y = 3(3) - 9 = 0. So (6sqrt(3), 0) is correct for t=-sqrt(3). For t = sqrt(3), x = (sqrt(3))^3 - 9(sqrt(3)) = 3sqrt(3) - 9sqrt(3) = -6sqrt(3), y = 0. So (-6sqrt(3), 0) is correct for t=sqrt(3). The labels in the comments say 't = -sqrt(3): vertical, at (6*sqrt(3), 0)' and 't = sqrt(3): vertical, at (-6*sqrt(3), 0)'. These are actually correct calculations. Wait, let me re-read carefully. Line 5 says 't = -sqrt(3): vertical, at (6*sqrt(3), 0)'. Line 6 says 't = sqrt(3): vertical, at (-6*sqrt(3), 0)'. The stated answer is 'vertical at (6*sqrt(3), 0), (-6*sqrt(3), 0)'. The solution finds the correct points. Why did I think it was wrong? Let me check the horizontal tangent. t=0, x=0, y=-9. Correct. Vertical tangents at t=±sqrt(3). Points are (±6sqrt(3), 0). The solution lists them correctly. Is there an error? Ah, look at line 5 and 6 comments. Line 5: 't = -sqrt(3): vertical, at (6*sqrt(3), 0)'. Line 6: 't = sqrt(3): vertical, at (-6*sqrt(3), 0)'. The calculations are correct. The verdict should be pass? Let me double check the problem statement. 'Find the points...'. The solution provides the points. The logic in line 3 is correct. The derivatives in 1 and 2 are correct. The substitution in 4 is correct. The substitutions in 5 and 6 are correct. The final answer matches. It seems correct. Wait, is there a subtle error? 'Horizontal tangent: dy/dt = 0 with dx/dt ≠ 0'. At t=0, dy/dt=0, dx/dt=-9≠0. Correct. 'Vertical: dx/dt = 0 with dy/dt ≠ 0'. At t=±sqrt(3), dx/dt=0, dy/dt=±6sqrt(3)≠0. Correct. The points are correct. I will mark it as pass.
  • gpt-oss:20b: fail (misleading) 2026-10-14 — The solution correctly identifies the parameter values for horizontal and vertical tangents, but it incorrectly uses the y‑formula (3*t**2-9) instead of the x‑formula (t**3-9*t) to compute the x‑coordinates of the vertical tangent points. Consequently, the justification for the points (±6√3,0) is incomplete and could mislead a student into thinking the calculation was performed correctly.

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.

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