Intermediate Value Theorem: locating a root
Problem 1.741 · medium
Show that \( \displaystyle x^{3} - 2 x = 0 \) has a solution in \( \displaystyle [-2, -1] \), then use two steps of bisection to find an interval of length \( \displaystyle \tfrac14 \) containing it.
- f is continuous on [-2, -1] (it is built from continuous functions), so the IVT applies.Reviewed
- \[ \left. x^{3} - 2 x \right|_{\substack{ x=-2 }} = -4 \]f(-2).✓ Proved
- \[ \left. x^{3} - 2 x \right|_{\substack{ x=-1 }} = 1 \]f(-1): the opposite sign, so a root lies between.✓ Proved
- \[ \left. x^{3} - 2 x \right|_{\substack{ x=- \frac{3}{2} }} = - \frac{3}{8} \]f(-3/2) ≈ -0.3750.✓ Proved
- \[ \left. x^{3} - 2 x \right|_{\substack{ x=- \frac{5}{4} }} = \frac{35}{64} \]f(-5/4) ≈ 0.5469.✓ Proved
- The sign changes on [-3/2, -5/4], so a root lies there.Reviewed
Answer \( \left[- \frac{3}{2}, - \frac{5}{4}\right] \)
Lines: 4 proved, 2 reviewed. The answer was also checked a second way, without looking at the solution. The reviewers disagree about how one step is explained; every verdict is in the receipt. 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 the specified interval.
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-10-21 — The solution correctly applies the Intermediate Value Theorem and performs two steps of bisection to isolate the root in the specified interval.gpt-oss:20b: pass 2026-10-21qwen3.6:27b-mlx: pass 2026-10-20 — The solution correctly applies the Intermediate Value Theorem and performs two steps of bisection to isolate the root in the specified interval.gpt-oss:20b: pass 2026-10-20qwen3.6:27b-mlx: fail (error) 2026-10-20 — The interval notation in the final answer is incorrect because the lower bound (-3/2) is greater than the upper bound (-5/4). Standard interval notation requires [a, b] with a <= b, so it should be [-5/4, -3/2] or the bounds should be swapped in the text.gpt-oss:20b: pass 2026-10-20
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 2026-10-21 with SymPy 1.14.0.