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Intermediate Value Theorem: locating a root

Problem 1.646 · medium

Show that \( \displaystyle x^{4} + x - 8 = 0 \) 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.
\[ \left. x^{4} + x - 8 \right|_{\substack{ x=-2 }} = 6 \]
f(-2).✓ Proved
\[ \left. x^{4} + x - 8 \right|_{\substack{ x=-1 }} = -8 \]
f(-1): the opposite sign, so a root lies between.✓ Proved
\[ \left. x^{4} + x - 8 \right|_{\substack{ x=- \frac{3}{2} }} = - \frac{71}{16} \]
f(-3/2) ≈ -4.4375.✓ Proved
\[ \left. x^{4} + x - 8 \right|_{\substack{ x=- \frac{7}{4} }} = - \frac{95}{256} \]
f(-7/4) ≈ -0.3711.✓ 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: pass — The solution omits an explicit sentence stating that the sign change at x=-3/2 guarantees a root in [-2,-3/2] before proceeding to the next bisection step. Adding this clarifies the two-step bisection process.
qwen3.6:27b-mlx: fail (error) — The solution fails to explicitly state that f(-2) > 0, which is required to apply the Intermediate Value Theorem alongside f(-1) < 0. Furthermore, the final interval [-2, -7/4] has length 1/4, but the problem asks for an interval containing the root; while the root is indeed in this interval, the standard bisection method would typically narrow down to [-7/4, -3/2] or similar depending on sign checks, but here the logic holds for existence. However, the primary error is the incomplete justification for the IVT application in step 1/3 (missing explicit sign of f(-2)).

Every verdict on record (6)

qwen3.6:27b-mlx: fail (error) 2026-10-18 — The solution fails to explicitly state that f(-2) > 0, which is required to apply the Intermediate Value Theorem alongside f(-1) < 0. Furthermore, the final interval [-2, -7/4] has length 1/4, but the problem asks for an interval containing the root; while the root is indeed in this interval, the standard bisection method would typically narrow down to [-7/4, -3/2] or similar depending on sign checks, but here the logic holds for existence. However, the primary error is the incomplete justification for the IVT application in step 1/3 (missing explicit sign of f(-2)).
gpt-oss:20b: pass 2026-10-18 — The solution omits an explicit sentence stating that the sign change at x=-3/2 guarantees a root in [-2,-3/2] before proceeding to the next bisection step. Adding this clarifies the two-step bisection process.
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: pass 2026-10-16
qwen3.6:27b-mlx: pass 2026-10-14 — 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: fail (misleading) 2026-10-14 — The solution skips the two explicit bisection steps and merely states that a sign change occurs on [-2,-7/4]; it does not justify that this interval results from two bisection iterations, so a student could be misled about the procedure.

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.