Home›Calculus 1›Increasing, decreasing and concavity›Problem 3.208

Increasing, decreasing and concavity

Problem 3.208 · medium

For \( \displaystyle f(x) = 2 x^{3} - 12 x^{2} + 18 x \), find the intervals where f is increasing or decreasing, where it is concave up or down, and its inflection points.

\[ \frac{d}{d x} \left(2 x^{3} - 12 x^{2} + 18 x\right) = \left(x - 1\right) \left(6 x - 18\right) \]
f' factored.✓ Proved
f' > 0 outside [1, 3] and f' < 0 between them (a positive quadratic with roots 1, 3).
Reviewed
\[ \frac{d^{2}}{d x^{2}} \left(2 x^{3} - 12 x^{2} + 18 x\right) = 12 x - 24 \]
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 \( \uparrow (-\infty,1) \cup (3,\infty);\ \downarrow (1,3);\ \text{concave down } (-\infty,2),\ \text{up } (2,\infty);\ \text{inflection at } x=2 \)

✓ 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: pass
qwen3.6:27b-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 2026-09-26
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 2026-09-26

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.