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Tangent lines

Problem 2.1038 · medium

Find an equation of the tangent line to \( \displaystyle y = - 4 x^{2} - 3 x - \frac{3}{x} \) at \( \displaystyle x = 1 \).

The tangent line passes through the point (a, f(a)) and has slope f'(a).
Reviewed
\[ \left. - 4 x^{2} - 3 x - \frac{3}{x} \right|_{\substack{ x=1 }} = -10 \]
The point of tangency.✓ Proved
\[ \frac{d}{d x} \left(- 4 x^{2} - 3 x - \frac{3}{x}\right) = - 8 x - 3 + \frac{3}{x^{2}} \]
Differentiate.✓ Proved
\[ \left. - 8 x - 3 + \frac{3}{x^{2}} \right|_{\substack{ x=1 }} = -8 \]
The slope at the point.✓ Proved
\[ - 8 x - 2 \]
Point-slope form, then simplify.✓ Proved

Answer \( y = - 8 x - 2 \)

Lines: 4 proved, 1 reviewed. 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.

The full receipt

Line	Status	Checked by	Detail
1	Reviewed	reviewers: gpt-oss:20b, qwen3.6:27b-mlx	a sentence; read, not computed
2	✓ Proved	sympy 1.14.0	simplify(lhs - rhs) reduced to 0
3	✓ Proved	sympy 1.14.0	simplify(lhs - rhs) reduced to 0
4	✓ Proved	sympy 1.14.0	simplify(lhs - rhs) reduced to 0
5	✓ 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	the line meets the curve at x = a, and its slope matches a central difference quotient of f there

Reviewers

gpt-oss:20b: pass
qwen3.6:27b-mlx: pass — The solution correctly identifies the point of tangency and the slope, and the final equation is algebraically correct.

Every verdict on record (4)

qwen3.6:27b-mlx: pass 2026-09-27 — The solution correctly identifies the point of tangency and the slope, and the final equation is algebraically correct.
gpt-oss:20b: pass 2026-09-27
qwen3.6:27b-mlx: fail (error) 2026-09-27 — The final equation is incorrect. The point-slope form should be y - (-10) = -8(x - 1), which simplifies to y = -8x - 2. However, the solution lists '-8*x - 2' as the result of simplification without explicitly showing the y = part or the intermediate step, and more critically, the stated answer in the problem description is 'Eq(y, -8*x - 2)', but the calculation in step 5 just outputs the RHS. Let's re-read carefully. Step 5 says '-8*x - 2, -8*x - 2'. This implies the equation is y = -8x - 2. Let's check the math. f(1) = -4 - 3 - 3 = -10. f'(1) = -8 - 3 + 3 = -8. Point-slope: y - (-10) = -8(x - 1) => y + 10 = -8x + 8 => y = -8x - 2. The math is correct. Why did I think it was wrong? Ah, the prompt says 'Stated answer: Eq(y, -8*x - 2)'. The solution step 5 is just an equation line '-8*x - 2, -8*x - 2'. This is not an equation for the line. It's just an identity. It fails to state 'y = ...'. The sentence in step 1 says 'The tangent line passes through... and has slope...'. Step 5 is labeled 'Point-slope form, then simplify'. But the equation shown is just 'RHS = RHS'. It does not show 'y = ...'. This is a severe formatting/logic error in the solution steps provided. It doesn't actually produce the equation of the line, just the expression for y. A student would not know to set y equal to it. It's misleading/error because it presents an identity as the result of simplifying the point-slope form, omitting the 'y =' part entirely.
gpt-oss:20b: pass 2026-09-27

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/tangent_line, checked 2026-09-27 with SymPy 1.14.0.