Absolute extrema on a closed interval
Problem 3.326 · hard
Find the absolute maximum and minimum values of \( \displaystyle f(x) = x^{3} + 3 x^{2} - 2 \) on \( \displaystyle [-2, 4] \).
- A continuous function on a closed interval has its extreme values at critical points or endpoints.
- \[ \frac{d}{d x} \left(x^{3} + 3 x^{2} - 2\right) = 3 x \left(x + 2\right) \]Differentiate and factor.✓ Proved
- Critical numbers inside [-2, 4]: 0.
- \[ \left. x^{3} + 3 x^{2} - 2 \right|_{\substack{ x=-2 }} = 2 \]f(-2).✓ Proved
- \[ \left. x^{3} + 3 x^{2} - 2 \right|_{\substack{ x=0 }} = -2 \]f(0).✓ Proved
- \[ \left. x^{3} + 3 x^{2} - 2 \right|_{\substack{ x=4 }} = 110 \]f(4).✓ Proved
- The largest value is 110 and the smallest is -2.
Answer \( \text{max } 110 \text{ at } x=4;\ \text{min } -2 \text{ at } x=0 \)
Lines: 4 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 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| 7 | 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 2026-09-29 — reviewer returned a non-objectgpt-oss:20b: pass 2026-09-29qwen3.6:27b-mlx: pass 2026-09-29 — The solution correctly identifies the critical points, evaluates the function at the critical points and endpoints, and correctly determines the absolute maximum and minimum values.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 2026-09-29 with SymPy 1.14.0.