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Absolute extrema on a closed interval

Problem 3.50 · hard Video

Find the absolute maximum and minimum values of \( \displaystyle f(x) = - x^{3} + 3 x^{2} + 9 x - 4 \) on \( \displaystyle [-4, 4] \).

A continuous function on a closed interval has its extreme values at critical points or endpoints.
Reviewed
\[ \frac{d}{d x} \left(- x^{3} + 3 x^{2} + 9 x - 4\right) = \left(9 - 3 x\right) \left(x + 1\right) \]
Differentiate and factor.✓ Proved
Critical numbers inside [-4, 4]: -1, 3.
Reviewed
\[ \left. - x^{3} + 3 x^{2} + 9 x - 4 \right|_{\substack{ x=-4 }} = 72 \]
f(-4).✓ Proved
\[ \left. - x^{3} + 3 x^{2} + 9 x - 4 \right|_{\substack{ x=-1 }} = -9 \]
f(-1).✓ Proved
\[ \left. - x^{3} + 3 x^{2} + 9 x - 4 \right|_{\substack{ x=3 }} = 23 \]
f(3).✓ Proved
\[ \left. - x^{3} + 3 x^{2} + 9 x - 4 \right|_{\substack{ x=4 }} = 16 \]
f(4).✓ Proved
The largest value is 72 and the smallest is -9.
Reviewed

Answer \( \text{max } 72 \text{ at } x=-4;\ \text{min } -9 \text{ at } x=-1 \)

The graph of f(x) = -x³ + 3x² + 9x - 4 on [-4, 4]. The candidates are the endpoints and the critical points inside; the highest is 72 and the lowest -9.

✓ Nihil obstat Lines: 5 proved, 3 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. 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	✓ Proved	sympy 1.14.0	simplify(lhs - rhs) reduced to 0
8	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

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-09-28
gpt-oss:20b: pass 2026-09-28
qwen3.6:27b-mlx: pass 2026-09-27
gpt-oss:20b: pass 2026-09-27
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/absolute_extrema, checked 2026-09-28 with SymPy 1.14.0.