Intermediate Value Theorem: locating a root
Problem 1.647 · medium
Show that \( \displaystyle x^{4} + x - 4 = 0 \) 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.
- \[ \left. x^{4} + x - 4 \right|_{\substack{ x=-2 }} = 10 \]f(-2).✓ Proved
- \[ \left. x^{4} + x - 4 \right|_{\substack{ x=-1 }} = -4 \]f(-1): the opposite sign, so a root lies between.✓ Proved
- \[ \left. x^{4} + x - 4 \right|_{\substack{ x=- \frac{3}{2} }} = - \frac{7}{16} \]f(-3/2) ≈ -0.4375.✓ Proved
- \[ \left. x^{4} + x - 4 \right|_{\substack{ x=- \frac{7}{4} }} = \frac{929}{256} \]f(-7/4) ≈ 3.6289.✓ Proved
- The sign changes on [-7/4, -3/2], so a root lies there.
Answer \( \left[- \frac{7}{4}, - \frac{3}{2}\right] \)
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: passqwen3.6:27b-mlx: fail (error) — The solution claims to perform 'two steps of bisection' but only performs one (evaluating the midpoint of the initial interval). A second step would require evaluating the midpoint of the new interval [-7/4, -3/2] to obtain an interval of length 1/4.
Every verdict on record (6)
qwen3.6:27b-mlx: fail (error) 2026-10-18 — The solution claims to perform 'two steps of bisection' but only performs one (evaluating the midpoint of the initial interval). A second step would require evaluating the midpoint of the new interval [-7/4, -3/2] to obtain an interval of length 1/4.gpt-oss:20b: pass 2026-10-18qwen3.6:27b-mlx: inconclusive 2026-10-16 — reviewer returned a non-objectgpt-oss:20b: pass 2026-10-16qwen3.6:27b-mlx: inconclusive 2026-10-14 — reviewer returned a non-objectgpt-oss:20b: pass 2026-10-14
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.