Which of the following statements is true? Bracketed methods with bounds [math: [a,b]] can only begin if:

Question

BlackTom AI · Accepted Answer

By the Intermediate Value Theorem, if a function f is continuous on [a,b] and f(a) and f(b) have opposite signs (i.e. f(a)·f(b) < 0), then there exists at least one c in (a,b) with f(c) = 0. This sign change of the function values at the endpoints is the basis for bracketing (interval) root-finding methods such as bisection.

1. Relevant knowledge points and concepts
- Continuity on [a,b] and the Intermediate Value Theorem (IVT).
- Bracketed root-finding methods require an interval known to contain a root.
- A sign change at the endpoints (f(a) and f(b) of opposite sign) is the standard condition guaranteeing a root inside the interval.

2. Complete reasoning process step by step
- Start with a continuous f on [a,b].
- Compute f(a) and f(b).
- If f(a)·f(b) < 0, IVT guarantees ∃c ∈ (a,b) s.t. f(c)=0; thus the interval brackets a root and bracketing methods can begin.
- If f(a)·f(b) > 0 (same sign) or equals 0 in a trivial endpoint-root case, the guarantee of an interior root is not given (endpoint could itself be a root if f(a)=0 or f(b)=0).

3. Calculations / intermediate steps (example)
Example: f(x)=x. Take a = −1, b = 2.
f(a) = −1, f(b) = 2, product = −2 < 0 → IVT guarantees a root. Indeed f(0)=0 lies in (−1,2).

4. Why the correct answer is correct
The correct answer states that "Function values at a and b are of opposite signs." That is precisely the f(a)·f(b) < 0 condition required by IVT to ensure a root lies between a and b, hence permitting bracketing methods.

5. Why the other options are incorrect
- a. "a and b are of the same sign": irrelevant — the signs of the endpoint x-values do not determine whether f has a root between them.
- b. "Function values at a and b are of the same sign": incorrect because same-sign function values do not guarantee a root inside (no IVT sign-change guarantee).
- c. "a and b are of opposite signs": also irrelevant — a and b being opposite in sign says nothing about f(a) and f(b); the function values might still be both positive or both negative.

6. Conclusion
Therefore, bracketing methods can only begin if the function values at a and b are of opposite signs, matching the provided correct answer.

Which of the following statements is true? Bracketed methods with bounds [math: [a,b]] can only begin if:单项选择题

类似问题

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

Which of these statements is true regarding the bisection method in solving \(f(x) = x^2 + x - 5\) with \(x_l= -3\) and \(x_u = 3\)?

We want to find a root to the equation [math: x3+x+1=0]x^3+x+1=0. Starting from the interval [math: [−1,0]][-1,0], apply the bisection method for [math: 3] times to obtain an approximation to the root. (Correct the answer to 2 decimal places.)

A project has a required return of 12.6 percent, an initial cash outflow of $42,100, and cash inflows of $16,500 in Year 1, $11,700 in Year 2, and $10,400 in Year 4. What is the net present value?

更多留学生实用工具

智能学习助手

风格化写作助手

论文查重助手

文献引用助手

课堂转译助手

课堂笔记助手

Quiz搜索助手

学校历年真题

智能刷题助手

智能匹配练习

希望你的学习变得更简单