Absolute extrema on a closed interval
Problem 3.47 · hard
Find the absolute maximum and minimum values of \( \displaystyle f(x) = x^{3} - 3 x \) on \( \displaystyle [-4, 3] \).
- A continuous function on a closed interval has its extreme values at critical points or endpoints.
- \[ \frac{d}{d x} \left(x^{3} - 3 x\right) = \left(x + 1\right) \left(3 x - 3\right) \]Differentiate and factor.✓ Proved
- Critical numbers inside [-4, 3]: -1, 1.
- \[ \left. x^{3} - 3 x \right|_{\substack{ x=-4 }} = -52 \]f(-4).✓ Proved
- \[ \left. x^{3} - 3 x \right|_{\substack{ x=-1 }} = 2 \]f(-1).✓ Proved
- \[ \left. x^{3} - 3 x \right|_{\substack{ x=1 }} = -2 \]f(1).✓ Proved
- \[ \left. x^{3} - 3 x \right|_{\substack{ x=3 }} = 18 \]f(3).✓ Proved
- The largest value is 18 and the smallest is -52.
Answer \( \text{max } 18 \text{ at } x=3;\ \text{min } -52 \text{ at } x=-4 \)
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.