Critical numbers
Problem 3.40 · easy
Find the critical numbers of \( \displaystyle f(x) = 3 x^{3} + 9 x^{2} - 72 x + 1 \).
- Critical numbers are where f'(x) = 0 or f' does not exist; this f' is a polynomial, so it exists everywhere.
- \[ \frac{d}{d x} \left(3 x^{3} + 9 x^{2} - 72 x + 1\right) = 9 x^{2} + 18 x - 72 \]Differentiate.✓ Proved
- \[ 9 x^{2} + 18 x - 72 = \left(x + 4\right) \left(9 x - 18\right) \]Factor f'(x).✓ Proved
- f'(x) = 0 at x = -4 and x = 2.
Answer \( x = -4, 2 \)
Lines: 2 proved, 2 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 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| 4 | Not checked | — | a sentence; read, not computed |
| answer, a second way | ✓ Checked independently | sympy 1.14.0 + mpmath 1.3.0 | f' changes sign exactly that many times on [-8, 8], sampled finely |
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/critical_numbers, checked 2026-09-26 with SymPy 1.14.0.