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Absolute extrema on a closed interval

Problem 3.324 · hard Video

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

A continuous function on a closed interval has its extreme values at critical points or endpoints.
Reviewed
\[ \frac{d}{d x} \left(x^{3} + \frac{3 x^{2}}{2} - 6 x\right) = \left(x + 2\right) \left(3 x - 3\right) \]
Differentiate and factor.✓ Proved
Critical numbers inside [-3, 2]: -2, 1.
Reviewed
\[ \left. x^{3} + \frac{3 x^{2}}{2} - 6 x \right|_{\substack{ x=-3 }} = \frac{9}{2} \]
f(-3).✓ Proved
\[ \left. x^{3} + \frac{3 x^{2}}{2} - 6 x \right|_{\substack{ x=-2 }} = 10 \]
f(-2).✓ Proved
\[ \left. x^{3} + \frac{3 x^{2}}{2} - 6 x \right|_{\substack{ x=1 }} = - \frac{7}{2} \]
f(1).✓ Proved
\[ \left. x^{3} + \frac{3 x^{2}}{2} - 6 x \right|_{\substack{ x=2 }} = 2 \]
f(2).✓ Proved
The largest value is 10 and the smallest is -7/2.
Reviewed

Answer \( \text{max } 10 \text{ at } x=-2;\ \text{min } - \frac{7}{2} \text{ at } x=1 \)

The graph of f(x) = x³ + 3x²/2 - 6x on [-3, 2]. The candidates are the endpoints and the critical points inside; the highest is 10 and the lowest -7/2.

✓ 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 2026-10-04
gpt-oss:20b: pass 2026-10-04
qwen3.6:27b-mlx: pass 2026-09-29
gpt-oss:20b: pass 2026-09-29
qwen3.6:27b-mlx: pass 2026-09-29
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.