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Absolute extrema on a closed interval

Problem 3.323 · medium Video

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

A continuous function on a closed interval has its extreme values at critical points or endpoints.
\[ \frac{d}{d x} \left(x^{3} + 3 x^{2} - 9 x - 3\right) = \left(x + 3\right) \left(3 x - 3\right) \]
Differentiate and factor.✓ Proved
Critical numbers inside [-3, 1]: none.
\[ \left. x^{3} + 3 x^{2} - 9 x - 3 \right|_{\substack{ x=-3 }} = 24 \]
f(-3).✓ Proved
\[ \left. x^{3} + 3 x^{2} - 9 x - 3 \right|_{\substack{ x=1 }} = -8 \]
f(1).✓ Proved
The largest value is 24 and the smallest is -8.

Answer \( \text{max } 24 \text{ at } x=-3;\ \text{min } -8 \text{ at } x=1 \)

The graph of f(x) = x³ + 3x² - 9x - 3 on [-3, 1]. The candidates are the endpoints and the critical points inside; the highest is 24 and the lowest -8.

Lines: 3 proved, 3 not checked. 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	Not checked	—	a sentence; read, not computed
2	✓ Proved	sympy 1.14.0	simplify(lhs - rhs) reduced to 0
3	Not checked	—	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	Not checked	—	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: inconclusive — reviewer returned a non-object

Every verdict on record (6)

qwen3.6:27b-mlx: inconclusive 2026-10-04 — reviewer returned a non-object
gpt-oss:20b: pass 2026-10-04
qwen3.6:27b-mlx: inconclusive 2026-09-29 — reviewer returned a non-object
gpt-oss:20b: pass 2026-09-29
qwen3.6:27b-mlx: inconclusive 2026-09-29 — reviewer returned a non-object
gpt-oss:20b: pass 2026-09-29

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 2026-10-04 with SymPy 1.14.0.