Which of the following statements applies to the bisection method used for finding the roots of functions?

Question

BlackTom AI · Accepted Answer

Bisection method relies on the Intermediate Value Theorem: if a function f is continuous on [a, b] and f(a) and f(b) have opposite signs (f(a)·f(b) < 0), then there exists at least one root in (a, b). The method repeatedly halves the interval where a sign change occurs to home in on a root.

Step-by-step reasoning:
1. Preconditions: f must be continuous on [a, b] and f(a)·f(b) < 0. These are the “requirements” that guarantee the method works.
2. Iteration: compute midpoint c = (a + b)/2. Evaluate f(c).
   - If f(c) = 0, c is the root.
   - If f(c)·f(a) < 0, replace b by c; else replace a by c.
3. Error reduction: each iteration halves the interval length. If initial interval length L0 = b − a, after n iterations the interval length Ln = L0 / 2^n. The maximum error (distance from midpoint to actual root) ≤ Ln/2 = L0 / 2^{n+1}.
4. To reach tolerance ε, need n ≥ log2(L0/ε) iterations (rounded up). Example: L0 = 1, ε = 10^{-6} ⇒ n ≥ log2(10^6) ≈ 19.93 ⇒ 20 iterations.

Why the correct answer is correct:
- Given the continuity and opposite-sign endpoints, the Intermediate Value Theorem guarantees at least one root in the interval and the bisection algorithm will converge to one of those roots. This is a rigorous guarantee when the requirements are met.

Why the other choices are incorrect:
- b. “Converges within a few iterations”: Not necessarily—precision demands may require many iterations (see log2 formula).
- c. “Is faster than Newton-Raphson”: False. Bisection converges linearly; Newton-Raphson typically has quadratic convergence near a simple root (faster) but requires derivative and good initial guess.
- d. “Requires a root to be at a turning point”: Incorrect. Bisection requires a sign change; a root at a turning point may not produce a sign change and can defeat the method’s precondition.

Conclusion: The correct option is a. Guaranteed to work for all continuous functions if requirements are met. This matches the provided answer.

Which of the following statements applies to the bisection method used for finding the roots of functions?单项选择题

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