Absolute extrema on a closed interval
Problem 3.51 · hard Video
Find the absolute maximum and minimum values of \( \displaystyle f(x) = x^{3} - 12 x + 2 \) on \( \displaystyle [-4, 3] \).
- A continuous function on a closed interval has its extreme values at critical points or endpoints.
- \[ \frac{d}{d x} \left(x^{3} - 12 x + 2\right) = \left(x + 2\right) \left(3 x - 6\right) \]Differentiate and factor.✓ Proved
- Critical numbers inside [-4, 3]: -2, 2.
- \[ \left. x^{3} - 12 x + 2 \right|_{\substack{ x=-4 }} = -14 \]f(-4).✓ Proved
- \[ \left. x^{3} - 12 x + 2 \right|_{\substack{ x=-2 }} = 18 \]f(-2).✓ Proved
- \[ \left. x^{3} - 12 x + 2 \right|_{\substack{ x=2 }} = -14 \]f(2).✓ Proved
- \[ \left. x^{3} - 12 x + 2 \right|_{\substack{ x=3 }} = -7 \]f(3).✓ Proved
- The largest value is 18 and the smallest is -14.
Answer \( \text{max } 18 \text{ at } x=-2;\ \text{min } -14 \text{ at } x=-4 \)
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 stated answer claims the minimum is at x = -4, but f(2) = -14 as well. The solution correctly calculates f(2) = -14 but fails to mention that the minimum occurs at both x = -4 and x = 2, making the stated answer incomplete/misleading regarding the location of the minimum.
Every verdict on record (6)
qwen3.6:27b-mlx: fail (error) 2026-09-28 — The stated answer claims the minimum is at x = -4, but f(2) = -14 as well. The solution correctly calculates f(2) = -14 but fails to mention that the minimum occurs at both x = -4 and x = 2, making the stated answer incomplete/misleading regarding the location of the minimum.gpt-oss:20b: pass 2026-09-28qwen3.6:27b-mlx: fail (error) 2026-09-27 — The stated answer claims the minimum is at x = -4, but the solution correctly calculates f(2) = -14 as well. Since f(-4) = f(2) = -14, the minimum occurs at both x = -4 and x = 2. The stated answer is incomplete/misleading by omitting x = 2.gpt-oss:20b: fail (misleading) 2026-09-27 — The solution states the minimum value is -14 at x = -4, but f(2) also equals -14, so the minimum occurs at two points. This omission could mislead a student into thinking -4 is the only minimizer.qwen3.6:27b-mlx: fail (error) 2026-09-26 — The stated answer claims the minimum is at x = -4, but f(2) = -14 as well. The solution correctly identifies -14 as the minimum value but fails to list x = 2 as a location for the minimum, making the stated answer incomplete and thus incorrect.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.