Absolute extrema on a closed interval
Problem 3.143 · medium Video
Find the absolute maximum and minimum values of \( \displaystyle f(x) = x^{3} - 3 x + 4 \) on \( \displaystyle [-1, 1] \).
- A continuous function on a closed interval has its extreme values at critical points or endpoints.
- \[ \frac{d}{d x} \left(x^{3} - 3 x + 4\right) = \left(x + 1\right) \left(3 x - 3\right) \]Differentiate and factor.✓ Proved
- Critical numbers inside [-1, 1]: none.
- \[ \left. x^{3} - 3 x + 4 \right|_{\substack{ x=-1 }} = 6 \]f(-1).✓ Proved
- \[ \left. x^{3} - 3 x + 4 \right|_{\substack{ x=1 }} = 2 \]f(1).✓ Proved
- The largest value is 6 and the smallest is 2.
Answer \( \text{max } 6 \text{ at } x=-1;\ \text{min } 2 \text{ at } x=1 \)
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. 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 | 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: inconclusive — reviewer returned a non-objectqwen3.6:27b-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: inconclusive 2026-09-28 — reviewer returned a non-objectqwen3.6:27b-mlx: inconclusive 2026-09-27 — reviewer returned a non-objectgpt-oss:20b: pass 2026-09-27qwen3.6:27b-mlx: pass 2026-09-26 — The solution correctly identifies the critical points, notes that they lie on the boundary of the interval (and thus are handled by the endpoint evaluation), and correctly compares the endpoint values to find the absolute extrema.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.