Absolute extrema on a closed interval
Problem 3.365 · medium
Find the absolute maximum and minimum values of \( \displaystyle f(x) = - x^{3} + \frac{3 x^{2}}{2} + 6 x - 4 \) on \( \displaystyle [-1, 2] \).
- A continuous function on a closed interval has its extreme values at critical points or endpoints.
- \[ \frac{d}{d x} \left(- x^{3} + \frac{3 x^{2}}{2} + 6 x - 4\right) = \left(6 - 3 x\right) \left(x + 1\right) \]Differentiate and factor.✓ Proved
- Critical numbers inside [-1, 2]: none.
- \[ \left. - x^{3} + \frac{3 x^{2}}{2} + 6 x - 4 \right|_{\substack{ x=-1 }} = - \frac{15}{2} \]f(-1).✓ Proved
- \[ \left. - x^{3} + \frac{3 x^{2}}{2} + 6 x - 4 \right|_{\substack{ x=2 }} = 6 \]f(2).✓ Proved
- The largest value is 6 and the smallest is -15/2.
Answer \( \text{max } 6 \text{ at } x=2;\ \text{min } - \frac{15}{2} \text{ at } x=-1 \)
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.
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: passqwen3.6:27b-mlx: inconclusive — reviewer returned a non-object
Every verdict on record (4)
qwen3.6:27b-mlx: inconclusive 2026-10-03 — reviewer returned a non-objectgpt-oss:20b: pass 2026-10-03qwen3.6:27b-mlx: pass 2026-10-03 — The solution correctly identifies that the critical points x = -1 and x = 2 lie on the boundary of the interval, meaning there are no critical numbers strictly inside the interval. The evaluation of endpoints and comparison of values is correct.gpt-oss:20b: inconclusive 2026-10-03 — reviewer returned a non-object
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 2026-10-03 with SymPy 1.14.0.