Increasing, decreasing and concavity
Problem 3.213 · medium
For \( \displaystyle f(x) = 2 x^{3} - 12 x^{2} \), find the intervals where f is increasing or decreasing, where it is concave up or down, and its inflection points.
- \[ \frac{d}{d x} \left(2 x^{3} - 12 x^{2}\right) = 6 x \left(x - 4\right) \]f' factored.✓ Proved
- f' > 0 outside [0, 4] and f' < 0 between them (a positive quadratic with roots 0, 4).Reviewed
- \[ \frac{d^{2}}{d x^{2}} \left(2 x^{3} - 12 x^{2}\right) = 12 x - 24 \]f''.✓ Proved
- \[ \left. 12 x - 24 \right|_{\substack{ x=2 }} = 0 \]f'' = 0 at x = 2, where it changes sign.✓ Proved
- f'' < 0 to the left of 2 (concave down) and f'' > 0 to the right (concave up).Reviewed
Answer \( \uparrow (-\infty,0) \cup (4,\infty);\ \downarrow (0,4);\ \text{concave down } (-\infty,2),\ \text{up } (2,\infty);\ \text{inflection at } x=2 \)
✓ Nihil obstat Lines: 3 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 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| 2 | Reviewed | reviewers: gpt-oss:20b, qwen3.6:27b-mlx | a sentence; read, not computed |
| 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 | 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 | the signs of f' and f'' were evaluated at a point inside each interval |
Reviewers
gpt-oss:20b: passqwen3.6:27b-mlx: pass — The solution correctly identifies the critical points and inflection point, and accurately determines the signs of the first and second derivatives to establish intervals of increase/decrease and concavity.
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-09-26 — The solution correctly identifies the critical points and inflection point, and accurately determines the signs of the first and second derivatives to establish intervals of increase/decrease and concavity.gpt-oss:20b: pass 2026-09-26qwen3.6:27b-mlx: pass 2026-09-26 — The solution correctly identifies the critical points and inflection point, and accurately determines the signs of the first and second derivatives to establish intervals of monotonicity and concavity.gpt-oss:20b: pass 2026-09-26
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/increasing_concavity, checked 2026-09-26 with SymPy 1.14.0.