Newton's method
Problem 3.412 · medium
Use Newton's method on \( \displaystyle f(x) = x^{3} - 2 x - 5 \) with \( \displaystyle x_0 = 2 \). Write the iteration \( \displaystyle x_{n+1} = F(x_n) \) and find \( \displaystyle x_1 \) and \( \displaystyle x_2 \) exactly.
- \[ \frac{d}{d x} \left(x^{3} - 2 x - 5\right) = 3 x^{2} - 2 \]f′(x).✓ Proved
- \[ x - \frac{x^{3} - 2 x - 5}{3 x^{2} - 2} = \frac{2 x^{3} + 5}{3 x^{2} - 2} \]Newton's formula x − f(x)/f′(x), simplified.✓ Proved
- \[ \left. \frac{2 x^{3} + 5}{3 x^{2} - 2} \right|_{\substack{ x=2 }} = \frac{21}{10} \]x₁ = F(x₀).✓ Proved
- \[ \left. \frac{2 x^{3} + 5}{3 x^{2} - 2} \right|_{\substack{ x=\frac{21}{10} }} = \frac{11761}{5615} \]x₂ = F(x₁).✓ Proved
Answer \( x_1 = \frac{21}{10},\ x_2 = \frac{11761}{5615} \approx 2.094568 \)
✓ Nihil obstat Every line of this solution was proved by the computer algebra system SymPy. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation.
The full receipt
| Line | Status | Checked by | Detail |
|---|---|---|---|
| 1 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| 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 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| answer, a second way | ✓ Checked independently | sympy 1.14.0 + mpmath 1.3.0 | two Newton steps in 30-digit floating point |
Reviewers
gpt-oss:20b: passqwen3.6:27b-mlx: pass — The solution correctly derives the Newton iteration function, simplifies it algebraically, and computes the exact values for x_1 and x_2 as requested.
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-10-04 — The solution correctly derives the Newton iteration function, simplifies it algebraically, and computes the exact values for x_1 and x_2 as requested.gpt-oss:20b: pass 2026-10-04qwen3.6:27b-mlx: pass 2026-10-04 — The solution correctly derives the Newton's method iteration formula and accurately computes the exact values for x_1 and x_2. The algebraic simplification and substitutions are correct.gpt-oss:20b: pass 2026-10-04
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/newtons_method, checked 2026-10-04 with SymPy 1.14.0.