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Absolute extrema on a closed interval

Problem 3.53 · hard Video

Find the absolute maximum and minimum values of \( \displaystyle f(x) = - x^{3} + \frac{9 x^{2}}{2} + 1 \) on \( \displaystyle [-2, 4] \).

A continuous function on a closed interval has its extreme values at critical points or endpoints.
\[ \frac{d}{d x} \left(- x^{3} + \frac{9 x^{2}}{2} + 1\right) = - 3 x \left(x - 3\right) \]
Differentiate and factor.✓ Proved
Critical numbers inside [-2, 4]: 0, 3.
\[ \left. - x^{3} + \frac{9 x^{2}}{2} + 1 \right|_{\substack{ x=-2 }} = 27 \]
f(-2).✓ Proved
\[ \left. - x^{3} + \frac{9 x^{2}}{2} + 1 \right|_{\substack{ x=0 }} = 1 \]
f(0).✓ Proved
\[ \left. - x^{3} + \frac{9 x^{2}}{2} + 1 \right|_{\substack{ x=3 }} = \frac{29}{2} \]
f(3).✓ Proved
\[ \left. - x^{3} + \frac{9 x^{2}}{2} + 1 \right|_{\substack{ x=4 }} = 9 \]
f(4).✓ Proved
The largest value is 27 and the smallest is 1.

Answer \( \text{max } 27 \text{ at } x=-2;\ \text{min } 1 \text{ at } x=0 \)

The graph of f(x) = -x³ + 9x²/2 + 1 on [-2, 4]. The candidates are the endpoints and the critical points inside; the highest is 27 and the lowest 1.

Lines: 5 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	✓ Proved	sympy 1.14.0	simplify(lhs - rhs) reduced to 0
7	✓ Proved	sympy 1.14.0	simplify(lhs - rhs) reduced to 0
8	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 2026-09-28 — reviewer returned a non-object
gpt-oss:20b: pass 2026-09-28
qwen3.6:27b-mlx: fail (error) 2026-09-27 — The solution incorrectly identifies 27 as the maximum value, ignoring the larger value 29/2 (14.5) found at x=3. It also fails to check the endpoint x=4 properly in the final comparison, although f(4)=9 is calculated.
gpt-oss:20b: pass 2026-09-27
qwen3.6:27b-mlx: fail (error) 2026-09-26 — The solution incorrectly identifies 27 as the maximum value, ignoring the larger value 29/2 (14.5) found at x=3. It also fails to check the endpoint x=4, although f(4)=9 is not the max or min, the omission of the true maximum at x=3 is the critical error.
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.