Increasing, decreasing and concavity
Problem 3.61 · medium
For \( \displaystyle f(x) = 3 x^{3} + 18 x^{2} + 27 x \), find the intervals where f is increasing or decreasing, where it is concave up or down, and its inflection points.
- \[ \frac{d}{d x} \left(3 x^{3} + 18 x^{2} + 27 x\right) = \left(x + 3\right) \left(9 x + 9\right) \]f' factored.✓ Proved
- f' > 0 outside [-3, -1] and f' < 0 between them (a positive quadratic with roots -3, -1).
- \[ \frac{d^{2}}{d x^{2}} \left(3 x^{3} + 18 x^{2} + 27 x\right) = 18 x + 36 \]f''.✓ Proved
- \[ \left. 18 x + 36 \right|_{\substack{ x=-2 }} = 0 \]f'' = 0 at x = -2, where it changes sign.✓ Proved
- f'' < 0 to the left of -2 (concave down) and f'' > 0 to the right (concave up).
Answer \( \uparrow (-\infty,-3) \cup (-1,\infty);\ \downarrow (-3,-1);\ \text{concave down } (-\infty,-2),\ \text{up } (-2,\infty);\ \text{inflection at } x=-2 \)
Lines: 3 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 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| 2 | Not checked | — | a sentence; read, not computed |
| 3 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| 4 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| 5 | Not checked | — | a sentence; read, not computed |
| answer, a second way | ✓ Checked independently | sympy 1.14.0 + mpmath 1.3.0 | the signs of f' and f'' were evaluated at a point inside each interval |
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/increasing_concavity, checked 2026-09-26 with SymPy 1.14.0.