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Absolute extrema on a closed interval

Problem 3.361 · hard

Find the absolute maximum and minimum values of \( \displaystyle f(x) = x^{3} - 3 x^{2} - 1 \) on \( \displaystyle [0, 3] \).

A continuous function on a closed interval has its extreme values at critical points or endpoints.
Reviewed
\[ \frac{d}{d x} \left(x^{3} - 3 x^{2} - 1\right) = 3 x \left(x - 2\right) \]
Differentiate and factor.✓ Proved
Critical numbers inside [0, 3]: 2.
Reviewed
\[ \left. x^{3} - 3 x^{2} - 1 \right|_{\substack{ x=0 }} = -1 \]
f(0).✓ Proved
\[ \left. x^{3} - 3 x^{2} - 1 \right|_{\substack{ x=2 }} = -5 \]
f(2).✓ Proved
\[ \left. x^{3} - 3 x^{2} - 1 \right|_{\substack{ x=3 }} = -1 \]
f(3).✓ Proved
The largest value is -1 and the smallest is -5.
Reviewed

Answer \( \text{max } -1 \text{ at } x=0;\ \text{min } -5 \text{ at } x=2 \)

Lines: 4 proved, 3 reviewed. The answer was also checked a second way, without looking at the solution. The reviewers disagree about how one step is explained; every verdict is in the receipt.

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	Reviewed	reviewers: gpt-oss:20b, qwen3.6:27b-mlx	a sentence; read, not computed
4	✓ Proved	sympy 1.14.0	simplify(lhs - rhs) reduced to 0
5	✓ Proved	sympy 1.14.0	simplify(lhs - rhs) reduced to 0
6	✓ Proved	sympy 1.14.0	simplify(lhs - rhs) reduced to 0
7	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 sampled at 40,001 points across the interval reaches the same max and min

Reviewers

gpt-oss:20b: pass
qwen3.6:27b-mlx: pass — The solution correctly identifies the critical points and endpoints, evaluates the function at these locations, and correctly determines the absolute maximum and minimum values.

Every verdict on record (4)

qwen3.6:27b-mlx: pass 2026-10-03 — The solution correctly identifies the critical points and endpoints, evaluates the function at these locations, and correctly determines the absolute maximum and minimum values.
gpt-oss:20b: pass 2026-10-03
qwen3.6:27b-mlx: fail (error) 2026-10-03 — The solution fails to identify x=0 as a critical point because f'(0)=0. While x=0 is an endpoint, omitting it from the list of critical numbers is mathematically incorrect and teaches a false definition of critical points.
gpt-oss:20b: pass 2026-10-03

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/absolute_extrema, checked 2026-10-03 with SymPy 1.14.0.