Absolute extrema on a closed interval
Problem 3.203 · medium
Find the absolute maximum and minimum values of \( \displaystyle f(x) = - x^{3} + \frac{3 x^{2}}{2} - 2 \) on \( \displaystyle [0, 1] \).
- A continuous function on a closed interval has its extreme values at critical points or endpoints.
- \[ \frac{d}{d x} \left(- x^{3} + \frac{3 x^{2}}{2} - 2\right) = - 3 x \left(x - 1\right) \]Differentiate and factor.✓ Proved
- Critical numbers inside [0, 1]: none.
- \[ \left. - x^{3} + \frac{3 x^{2}}{2} - 2 \right|_{\substack{ x=0 }} = -2 \]f(0).✓ Proved
- \[ \left. - x^{3} + \frac{3 x^{2}}{2} - 2 \right|_{\substack{ x=1 }} = - \frac{3}{2} \]f(1).✓ Proved
- The largest value is -3/2 and the smallest is -2.
Answer \( \text{max } - \frac{3}{2} \text{ at } x=1;\ \text{min } -2 \text{ at } x=0 \)
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 2026-09-26 — reviewer returned a non-objectgpt-oss:20b: pass 2026-09-26qwen3.6:27b-mlx: inconclusive 2026-09-26 — reviewer returned a non-objectgpt-oss:20b: inconclusive 2026-09-26 — 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 2026-09-26 with SymPy 1.14.0.