Consider the equation $y=x^{2}$ which has a root at $x=0$. Which of the following is true?

Question

BlackTom AI · Accepted Answer

Because y = x^2 ≥ 0 for all real x, the function never takes opposite signs on any interval that does not already include the root as an endpoint. Let f(x) = x^2. For any two trial endpoints a and b:

f(a) f(b) = a^2 b^2 ≥ 0,

and if neither a nor b equals 0 then a^2 b^2 > 0, so f(a) and f(b) have the same sign (both positive). Both the bisection and the false position (regula falsi) methods require a bracket [a,b] with f(a)·f(b) < 0 (a sign change) to proceed iteratively and guarantee that a root lies between a and b by the Intermediate Value Theorem. Without a sign change the methods cannot compute the midpoint or secant that is guaranteed to bracket the root.

Two edge cases to note:
- If one endpoint is chosen exactly at x = 0, then f(endpoint) = 0 and you have already found the root immediately (no iteration needed). This is not the same as the methods discovering an interior root by bracketing; it requires prior knowledge or guessing the root as an endpoint.
- Because x = 0 is a double (even-multiplicity) root, f does not change sign across the root, so any bracket strictly containing 0 with nonzero endpoints will have f(a) and f(b) both positive, preventing the usual bracketing step.

Why the multiple-choice options are incorrect:
- (a) Both bisection and false position can determine the root: incorrect because both require a sign change; none exists for x^2 unless the root is chosen as an endpoint.
- (b) Only bisection can determine the root: incorrect; bisection has the same sign-change requirement as false position.
- (c) Only false position can determine the root: incorrect for the same reason.
- (d) Neither bisection or false position can determine the root: correct, since no sign change occurs for x^2 on any interval with nonzero endpoints, so neither method can find the interior root by bracketing.

Therefore the correct answer is: Neither bisection or false position can determine the root.

Consider the equation \(y=x^{2}\) which has a root at \(x=0\). Which of the following is true?单项选择题

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