Absolute extrema on a closed interval
Problem 3.137 · hard
Find the absolute maximum and minimum values of \( \displaystyle f(x) = - x^{3} - 6 x^{2} - 9 x - 1 \) on \( \displaystyle [-4, 2] \).
- A continuous function on a closed interval has its extreme values at critical points or endpoints.
- \[ \frac{d}{d x} \left(- x^{3} - 6 x^{2} - 9 x - 1\right) = \left(- 3 x - 3\right) \left(x + 3\right) \]Differentiate and factor.✓ Proved
- Critical numbers inside [-4, 2]: -3, -1.
- \[ \left. - x^{3} - 6 x^{2} - 9 x - 1 \right|_{\substack{ x=-4 }} = 3 \]f(-4).✓ Proved
- \[ \left. - x^{3} - 6 x^{2} - 9 x - 1 \right|_{\substack{ x=-3 }} = -1 \]f(-3).✓ Proved
- \[ \left. - x^{3} - 6 x^{2} - 9 x - 1 \right|_{\substack{ x=-1 }} = 3 \]f(-1).✓ Proved
- \[ \left. - x^{3} - 6 x^{2} - 9 x - 1 \right|_{\substack{ x=2 }} = -51 \]f(2).✓ Proved
- The largest value is 3 and the smallest is -51.
Answer \( \text{max } 3 \text{ at } x=-4;\ \text{min } -51 \text{ at } x=2 \)
Lines: 5 proved, 3 not checked. The answer was also checked a second way, without looking at the solution. The explanation has not been reviewed yet.
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 |
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.