Absolute extrema on a closed interval
Problem 3.145 · hard Video
Find the absolute maximum and minimum values of \( \displaystyle f(x) = - x^{3} - 3 x^{2} + 9 x - 2 \) on \( \displaystyle [-4, 4] \).
- A continuous function on a closed interval has its extreme values at critical points or endpoints.
- \[ \frac{d}{d x} \left(- x^{3} - 3 x^{2} + 9 x - 2\right) = \left(3 - 3 x\right) \left(x + 3\right) \]Differentiate and factor.✓ Proved
- Critical numbers inside [-4, 4]: -3, 1.
- \[ \left. - x^{3} - 3 x^{2} + 9 x - 2 \right|_{\substack{ x=-4 }} = -22 \]f(-4).✓ Proved
- \[ \left. - x^{3} - 3 x^{2} + 9 x - 2 \right|_{\substack{ x=-3 }} = -29 \]f(-3).✓ Proved
- \[ \left. - x^{3} - 3 x^{2} + 9 x - 2 \right|_{\substack{ x=1 }} = 3 \]f(1).✓ Proved
- \[ \left. - x^{3} - 3 x^{2} + 9 x - 2 \right|_{\substack{ x=4 }} = -78 \]f(4).✓ Proved
- The largest value is 3 and the smallest is -78.
Answer \( \text{max } 3 \text{ at } x=1;\ \text{min } -78 \text{ at } x=4 \)
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: passqwen3.6:27b-mlx: fail (error) — The solution fails to evaluate the function at the left endpoint x = -4, which yields f(-4) = 22. Since 22 > 3, the absolute maximum is 22, not 3. The omission of this endpoint evaluation leads to an incorrect final answer.
Every verdict on record (6)
qwen3.6:27b-mlx: fail (error) 2026-09-28 — The solution fails to evaluate the function at the left endpoint x = -4, which yields f(-4) = 22. Since 22 > 3, the absolute maximum is 22, not 3. The omission of this endpoint evaluation leads to an incorrect final answer.gpt-oss:20b: pass 2026-09-28qwen3.6:27b-mlx: fail (error) 2026-09-27 — The solution fails to evaluate the function at the left endpoint x = -4, which yields -22. While -22 is not the global minimum, omitting an endpoint evaluation is a procedural error in the Extreme Value Theorem application. Additionally, the stated answer claims the minimum is at x=4, but the solution text does not explicitly compare f(-4) to f(4) to justify ignoring -4 for the minimum, though the final numbers are correct. The primary error is the omission of the endpoint evaluation step.gpt-oss:20b: pass 2026-09-27qwen3.6:27b-mlx: fail (error) 2026-09-26 — The solution fails to evaluate the function at the left endpoint x = -4, which yields f(-4) = 22. Since 22 > 3, the absolute maximum is 22, not 3. The conclusion is incorrect because the comparison set was incomplete.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.