Critical numbers
Problem 3.233 · easy
Find the critical numbers of \( \displaystyle f(x) = - 3 x^{3} + 9 x^{2} - 2 \).
- Critical numbers are where f'(x) = 0 or f' does not exist; this f' is a polynomial, so it exists everywhere.Reviewed
- \[ \frac{d}{d x} \left(- 3 x^{3} + 9 x^{2} - 2\right) = - 9 x^{2} + 18 x \]Differentiate.✓ Proved
- \[ - 9 x^{2} + 18 x = - 9 x \left(x - 2\right) \]Factor f'(x).✓ Proved
- f'(x) = 0 at x = 0 and x = 2.Reviewed
Answer \( x = 0, 2 \)
✓ 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: passqwen3.6:27b-mlx: pass
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-09-27gpt-oss:20b: pass 2026-09-27qwen3.6:27b-mlx: pass 2026-09-27gpt-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.