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Critical numbers

Problem 3.232 · easy

Find the critical numbers of \( \displaystyle f(x) = 2 x^{3} + 9 x^{2} + 12 x - 1 \).

Critical numbers are where f'(x) = 0 or f' does not exist; this f' is a polynomial, so it exists everywhere.
Reviewed
\[ \frac{d}{d x} \left(2 x^{3} + 9 x^{2} + 12 x - 1\right) = 6 x^{2} + 18 x + 12 \]
Differentiate.✓ Proved
\[ 6 x^{2} + 18 x + 12 = \left(x + 2\right) \left(6 x + 6\right) \]
Factor f'(x).✓ Proved
f'(x) = 0 at x = -2 and x = -1.
Reviewed

Answer \( x = -2, -1 \)

✓ Nihil obstat Lines: 2 proved, 2 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation.

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	Reviewed	reviewers: gpt-oss:20b, qwen3.6:27b-mlx	a sentence; read, not computed
answer, a second way	✓ Checked independently	sympy 1.14.0 + mpmath 1.3.0	f' changes sign exactly that many times on [-8, 8], sampled finely

Reviewers

gpt-oss:20b: pass
qwen3.6:27b-mlx: pass

Every verdict on record (4)

qwen3.6:27b-mlx: pass 2026-09-27
gpt-oss:20b: pass 2026-09-27
qwen3.6:27b-mlx: pass 2026-09-27 — The solution correctly identifies the definition of critical numbers, computes the derivative, factors it, and solves for the roots. The logic is sound and complete.
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/critical_numbers, checked 2026-09-27 with SymPy 1.14.0.