Home›Calculus 1›Increasing, decreasing and concavity›Problem 3.211

Increasing, decreasing and concavity

Problem 3.211 · medium

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

\[ \frac{d}{d x} \left(x^{3} - 3 x^{2}\right) = 3 x \left(x - 2\right) \]
f' factored.✓ Proved
f' > 0 outside [0, 2] and f' < 0 between them (a positive quadratic with roots 0, 2).
Reviewed
\[ \frac{d^{2}}{d x^{2}} \left(x^{3} - 3 x^{2}\right) = 6 x - 6 \]
f''.✓ Proved
\[ \left. 6 x - 6 \right|_{\substack{ x=1 }} = 0 \]
f'' = 0 at x = 1, where it changes sign.✓ Proved
f'' < 0 to the left of 1 (concave down) and f'' > 0 to the right (concave up).
Reviewed

Answer \( \uparrow (-\infty,0) \cup (2,\infty);\ \downarrow (0,2);\ \text{concave down } (-\infty,1),\ \text{up } (1,\infty);\ \text{inflection at } x=1 \)

✓ 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 sign of the derivatives in the relevant intervals to establish monotonicity 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 sign of the derivatives in the relevant intervals to establish monotonicity and concavity.
gpt-oss:20b: pass 2026-09-26
qwen3.6:27b-mlx: pass 2026-09-26
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.