Intermediate Value Theorem: locating a root
Problem 1.684 · medium
Show that \( \displaystyle x^{3} - x + 4 = 0 \) 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.
- \[ \left. x^{3} - x + 4 \right|_{\substack{ x=-2 }} = -2 \]f(-2).✓ Proved
- \[ \left. x^{3} - x + 4 \right|_{\substack{ x=-1 }} = 4 \]f(-1): the opposite sign, so a root lies between.✓ Proved
- \[ \left. x^{3} - x + 4 \right|_{\substack{ x=- \frac{3}{2} }} = \frac{17}{8} \]f(-3/2) ≈ 2.1250.✓ Proved
- \[ \left. x^{3} - x + 4 \right|_{\substack{ x=- \frac{7}{4} }} = \frac{25}{64} \]f(-7/4) ≈ 0.3906.✓ Proved
- The sign changes on [-2, -7/4], so a root lies there.
Answer \( \left[-2, - \frac{7}{4}\right] \)
Lines: 4 proved, 2 not checked. 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 | 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 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| 5 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| 6 | Not checked | — | 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: fail (misleading) — The solution skips the explicit bisection steps: it computes f(-3/2) and f(-7/4) but does not state that f(-3/2)>0 implies the root lies in [-2,-3/2], nor that f(-7/4)>0 further narrows it to [-2,-7/4]. The conclusion about the interval is not justified by the preceding sentences, which could mislead a student about the bisection procedure.qwen3.6:27b-mlx: pass — The solution correctly applies the Intermediate Value Theorem and performs two steps of bisection. The final interval [-2, -7/4] has length 1/4 and contains a sign change, satisfying all problem requirements.
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-10-18 — The solution correctly applies the Intermediate Value Theorem and performs two steps of bisection. The final interval [-2, -7/4] has length 1/4 and contains a sign change, satisfying all problem requirements.gpt-oss:20b: fail (misleading) 2026-10-18 — The solution skips the explicit bisection steps: it computes f(-3/2) and f(-7/4) but does not state that f(-3/2)>0 implies the root lies in [-2,-3/2], nor that f(-7/4)>0 further narrows it to [-2,-7/4]. The conclusion about the interval is not justified by the preceding sentences, which could mislead a student about the bisection procedure.qwen3.6:27b-mlx: pass 2026-10-16 — The solution correctly applies the Intermediate Value Theorem and performs two steps of bisection to isolate the root in the interval [-2, -7/4], which has the required length of 1/4.gpt-oss:20b: fail (misleading) 2026-10-16 — The solution claims to use two steps of bisection but only performs one midpoint test; it does not actually carry out the second bisection step to reach an interval of length 1/4.qwen3.6:27b-mlx: pass 2026-10-15 — The solution correctly applies the Intermediate Value Theorem and performs two steps of bisection, arriving at the correct interval [-2, -7/4] of length 1/4.gpt-oss:20b: pass 2026-10-15
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 2026-10-18 with SymPy 1.14.0.