Intermediate Value Theorem: locating a root
Problem 1.823 · medium
Show that \( \displaystyle x^{3} - 3 x - 6 = 0 \) has a solution in \( \displaystyle [2, 3] \), then use two steps of bisection to find an interval of length \( \displaystyle \tfrac14 \) containing it.
- f is continuous on [2, 3] (it is built from continuous functions), so the IVT applies.Reviewed
- \[ \left. x^{3} - 3 x - 6 \right|_{\substack{ x=2 }} = -4 \]f(2).✓ Proved
- \[ \left. x^{3} - 3 x - 6 \right|_{\substack{ x=3 }} = 12 \]f(3): the opposite sign, so a root lies between.✓ Proved
- \[ \left. x^{3} - 3 x - 6 \right|_{\substack{ x=\frac{5}{2} }} = \frac{17}{8} \]f(5/2) ≈ 2.1250.✓ Proved
- \[ \left. x^{3} - 3 x - 6 \right|_{\substack{ x=\frac{9}{4} }} = - \frac{87}{64} \]f(9/4) ≈ -1.3594.✓ Proved
- The sign changes on [9/4, 5/2], so a root lies there.Reviewed
Answer \( \left[\frac{9}{4}, \frac{5}{2}\right] \)
✓ Nihil obstat Lines: 4 proved, 2 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. How we verify
The full receipt
| Line | Status | Checked by | Detail |
|---|---|---|---|
| 1 | Reviewed | reviewers: gpt-oss:20b, qwen3.6:27b-mlx | 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 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| 5 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| 6 | Reviewed | reviewers: gpt-oss:20b, qwen3.6:27b-mlx | a sentence; read, not computed |
| answer, a second way | ✓ Checked independently | sympy 1.14.0 + mpmath 1.3.0 | mpmath's bisection finds the root inside the claimed interval |
Reviewers
gpt-oss:20b: passqwen3.6:27b-mlx: pass — The solution correctly applies the Intermediate Value Theorem and performs two steps of bisection to isolate the root in an interval of length 1/4. The logic and calculations are sound.
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-11-12 — The solution correctly applies the Intermediate Value Theorem and performs two steps of bisection to isolate the root in an interval of length 1/4. The logic and calculations are sound.gpt-oss:20b: pass 2026-11-12qwen3.6:27b-mlx: pass 2026-11-11 — The solution correctly applies the Intermediate Value Theorem and performs two steps of bisection to isolate the root in the interval [9/4, 5/2], which has the required length of 1/4.gpt-oss:20b: pass 2026-11-11qwen3.6:27b-mlx: pass 2026-11-11 — The solution correctly applies the Intermediate Value Theorem and performs two steps of bisection, resulting in the correct interval of length 1/4.gpt-oss:20b: pass 2026-11-11
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/ivt_root, checked 2026-11-12 with SymPy 1.14.0.