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Intermediate Value Theorem: locating a root

Problem 1.747 · medium

Show that \( \displaystyle x^{3} + 3 x + 2 = 0 \) has a solution in \( \displaystyle [-1, 0] \), then use two steps of bisection to find an interval of length \( \displaystyle \tfrac14 \) containing it.

f is continuous on [-1, 0] (it is built from continuous functions), so the IVT applies.
Reviewed
\[ \left. x^{3} + 3 x + 2 \right|_{\substack{ x=-1 }} = -2 \]
f(-1).✓ Proved
\[ \left. x^{3} + 3 x + 2 \right|_{\substack{ x=0 }} = 2 \]
f(0): the opposite sign, so a root lies between.✓ Proved
\[ \left. x^{3} + 3 x + 2 \right|_{\substack{ x=- \frac{1}{2} }} = \frac{3}{8} \]
f(-1/2) ≈ 0.3750.✓ Proved
\[ \left. x^{3} + 3 x + 2 \right|_{\substack{ x=- \frac{3}{4} }} = - \frac{43}{64} \]
f(-3/4) ≈ -0.6719.✓ Proved
The sign changes on [-3/4, -1/2], so a root lies there.
Reviewed

Answer \( \left[- \frac{3}{4}, - \frac{1}{2}\right] \)

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: pass
qwen3.6:27b-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.

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 an interval of length 1/4.
gpt-oss:20b: pass 2026-10-21
qwen3.6:27b-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 2026-10-20
qwen3.6:27b-mlx: inconclusive 2026-10-20 — reviewer returned a non-object
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.