Increasing, decreasing and concavity
Problem 3.249 · medium
For \( \displaystyle f(x) = 3 x^{3} + \frac{9 x^{2}}{2} - 18 x \), find the intervals where f is increasing or decreasing, where it is concave up or down, and its inflection points.
- \[ \frac{d}{d x} \left(3 x^{3} + \frac{9 x^{2}}{2} - 18 x\right) = \left(x + 2\right) \left(9 x - 9\right) \]f' factored.✓ Proved
- f' > 0 outside [-2, 1] and f' < 0 between them (a positive quadratic with roots -2, 1).Reviewed
- \[ \frac{d^{2}}{d x^{2}} \left(3 x^{3} + \frac{9 x^{2}}{2} - 18 x\right) = 18 x + 9 \]f''.✓ Proved
- \[ \left. 18 x + 9 \right|_{\substack{ x=- \frac{1}{2} }} = 0 \]f'' = 0 at x = - \frac{1}{2}, where it changes sign.✓ Proved
- f'' < 0 to the left of - \frac{1}{2} (concave down) and f'' > 0 to the right (concave up).Reviewed
Answer \( \uparrow (-\infty,-2) \cup (1,\infty);\ \downarrow (-2,1);\ \text{concave down } (-\infty,- \frac{1}{2}),\ \text{up } (- \frac{1}{2},\infty);\ \text{inflection at } x=- \frac{1}{2} \)
✓ 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 2026-09-27 — 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 2026-09-27qwen3.6:27b-mlx: pass 2026-09-27 — The solution correctly identifies the critical points and inflection point, and accurately determines the sign of the derivatives in the relevant intervals. The reasoning is sound and the conclusions match the stated answer.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/increasing_concavity, checked 2026-09-27 with SymPy 1.14.0.