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Absolute extrema on a closed interval

Problem 3.205 · hard Video

Find the absolute maximum and minimum values of \( \displaystyle f(x) = x^{3} - 6 x^{2} + 9 x + 3 \) on \( \displaystyle [-1, 3] \).

A continuous function on a closed interval has its extreme values at critical points or endpoints.
Reviewed
\[ \frac{d}{d x} \left(x^{3} - 6 x^{2} + 9 x + 3\right) = \left(x - 1\right) \left(3 x - 9\right) \]
Differentiate and factor.✓ Proved
Critical numbers inside [-1, 3]: 1.
Reviewed
\[ \left. x^{3} - 6 x^{2} + 9 x + 3 \right|_{\substack{ x=-1 }} = -13 \]
f(-1).✓ Proved
\[ \left. x^{3} - 6 x^{2} + 9 x + 3 \right|_{\substack{ x=1 }} = 7 \]
f(1).✓ Proved
\[ \left. x^{3} - 6 x^{2} + 9 x + 3 \right|_{\substack{ x=3 }} = 3 \]
f(3).✓ Proved
The largest value is 7 and the smallest is -13.
Reviewed

Answer \( \text{max } 7 \text{ at } x=1;\ \text{min } -13 \text{ at } x=-1 \)

The graph of f(x) = x³ - 6x² + 9x + 3 on [-1, 3]. The candidates are the endpoints and the critical points inside; the highest is 7 and the lowest -13.

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. How we verify

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, evaluates the function at the critical points and endpoints, and correctly determines the absolute maximum and minimum values.

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-09-27 — The solution correctly identifies the critical points, evaluates the function at the critical points and endpoints, and correctly determines the absolute maximum and minimum values.
gpt-oss:20b: pass 2026-09-27
qwen3.6:27b-mlx: inconclusive 2026-09-26 — reviewer returned a non-object
gpt-oss:20b: pass 2026-09-26
qwen3.6:27b-mlx: inconclusive 2026-09-26 — reviewer returned a non-object
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/absolute_extrema, checked 2026-09-27 with SymPy 1.14.0.