Absolute extrema on a closed interval
Problem 3.141 · hard Video
Find the absolute maximum and minimum values of \( \displaystyle f(x) = - x^{3} + 12 x + 1 \) on \( \displaystyle [-2, 4] \).
- A continuous function on a closed interval has its extreme values at critical points or endpoints.
- \[ \frac{d}{d x} \left(- x^{3} + 12 x + 1\right) = \left(6 - 3 x\right) \left(x + 2\right) \]Differentiate and factor.✓ Proved
- Critical numbers inside [-2, 4]: 2.
- \[ \left. - x^{3} + 12 x + 1 \right|_{\substack{ x=-2 }} = -15 \]f(-2).✓ Proved
- \[ \left. - x^{3} + 12 x + 1 \right|_{\substack{ x=2 }} = 17 \]f(2).✓ Proved
- \[ \left. - x^{3} + 12 x + 1 \right|_{\substack{ x=4 }} = -15 \]f(4).✓ Proved
- The largest value is 17 and the smallest is -15.
Answer \( \text{max } 17 \text{ at } x=2;\ \text{min } -15 \text{ at } x=-2 \)
Lines: 4 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 | 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: passqwen3.6:27b-mlx: inconclusive — reviewer returned a non-object
Every verdict on record (6)
qwen3.6:27b-mlx: inconclusive 2026-09-28 — reviewer returned a non-objectgpt-oss:20b: pass 2026-09-28qwen3.6:27b-mlx: inconclusive 2026-09-27 — reviewer returned a non-objectgpt-oss:20b: pass 2026-09-27qwen3.6:27b-mlx: fail (error) 2026-09-26 — The solution fails to evaluate the function at the endpoint x=4, which yields -15. While the final minimum value is correct, the reasoning is incomplete because it ignores a boundary point that achieves the minimum, potentially misleading students about the necessity of checking all endpoints.gpt-oss:20b: fail (misleading) 2026-09-26 — The solution states the minimum occurs only at x = –2, but f(4) also equals –15, so the conclusion about the location of the minimum is incomplete.
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.