Absolute extrema on a closed interval
Problem 3.319 · hard Video
Find the absolute maximum and minimum values of \( \displaystyle f(x) = x^{3} + 3 x^{2} - 9 x \) on \( \displaystyle [-4, 3] \).
- A continuous function on a closed interval has its extreme values at critical points or endpoints.
- \[ \frac{d}{d x} \left(x^{3} + 3 x^{2} - 9 x\right) = \left(x + 3\right) \left(3 x - 3\right) \]Differentiate and factor.✓ Proved
- Critical numbers inside [-4, 3]: -3, 1.
- \[ \left. x^{3} + 3 x^{2} - 9 x \right|_{\substack{ x=-4 }} = 20 \]f(-4).✓ Proved
- \[ \left. x^{3} + 3 x^{2} - 9 x \right|_{\substack{ x=-3 }} = 27 \]f(-3).✓ Proved
- \[ \left. x^{3} + 3 x^{2} - 9 x \right|_{\substack{ x=1 }} = -5 \]f(1).✓ Proved
- \[ \left. x^{3} + 3 x^{2} - 9 x \right|_{\substack{ x=3 }} = 27 \]f(3).✓ Proved
- The largest value is 27 and the smallest is -5.
Answer \( \text{max } 27 \text{ at } x=-3;\ \text{min } -5 \text{ at } x=1 \)
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 maximum is at x=-3, but the solution correctly calculates f(3)=27, meaning the maximum also occurs at x=3. The final sentence fails to report this second location, making the stated answer incomplete and inconsistent with the work shown.
Every verdict on record (6)
qwen3.6:27b-mlx: fail (error) 2026-10-04 — The stated answer claims the maximum is at x=-3, but the solution correctly calculates f(3)=27, meaning the maximum also occurs at x=3. The final sentence fails to report this second location, making the stated answer incomplete and inconsistent with the work shown.gpt-oss:20b: pass 2026-10-04qwen3.6:27b-mlx: fail (misleading) 2026-09-29 — The solution claims the maximum is 27 at x = -3, but f(3) is also 27. By omitting x = 3, the solution incorrectly implies the maximum occurs only at x = -3, which is factually incomplete and misleading.gpt-oss:20b: pass 2026-09-29qwen3.6:27b-mlx: fail (error) 2026-09-29 — The stated answer claims the maximum is at x = -3, but f(3) is also 27. The solution correctly calculates f(3) = 27 in step 7 but fails to include x = 3 in the final conclusion, omitting a location where the absolute maximum occurs.gpt-oss:20b: pass 2026-09-29
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 2026-10-04 with SymPy 1.14.0.