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Critical numbers

Problem 3.236 · easy

Find the critical numbers of \( \displaystyle f(x) = 2 x^{3} + 27 x^{2} + 120 x + 2 \).

Critical numbers are where f'(x) = 0 or f' does not exist; this f' is a polynomial, so it exists everywhere.
Reviewed
\[ \frac{d}{d x} \left(2 x^{3} + 27 x^{2} + 120 x + 2\right) = 6 x^{2} + 54 x + 120 \]
Differentiate.✓ Proved
\[ 6 x^{2} + 54 x + 120 = \left(x + 5\right) \left(6 x + 24\right) \]
Factor f'(x).✓ Proved
f'(x) = 0 at x = -5 and x = -4.
Reviewed

Answer \( x = -5, -4 \)

✓ 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 — 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.

Every verdict on record (4)

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
qwen3.6:27b-mlx: pass 2026-09-27
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.