The graph of
𝑦
=
𝑓
(
𝑥
)
is given below.

If Newton's method is applied to the function
𝑓
, starting with an initial guess of
𝑥
1
=
1
, what happens to the iterates
𝑥
2
,
𝑥
3
,
𝑥
4
,
…
?

Question

The graph of 
𝑦
=
𝑓
(
𝑥
)
  is given below.

If Newton's method is applied to the function 
𝑓
 , starting with an initial guess of 
𝑥
1
=
1
 , what happens to the iterates 
𝑥
2
,
𝑥
3
,
𝑥
4
,
…
 ?

BlackTom AI · Accepted Answer

To analyze the Newton iteration sequence, we start from the information given: the initial guess is x1 = 1, and we’re applying Newton's method to the function f with the goal of locating a root, presumably at x = 5 from the answer choices.

Option A: "x2 = 1.5, x3 = 3, and x4 = 6.75."
- This option provides explicit iterates. It begins at x1 = 1 and proposes x2 = 1.5, which is a modest move toward 5. The next iterate x3 = 3 is closer to 5 than 1.5 in terms of absolute distance, but the following x4 = 6.75 overshoots past 5 and lies farther from 5 than x3 did. In Newton's method, such overshooting is not forbidden and can occur, especially if f is such that the tangent at these points produces large steps. However, for a sequence to be convergent to the root at x = 5, we typically expect the iterates to approach 5 (either monotonically or with diminishing oscillations) rather than bounce back and forth with increasing or irregular spacing. The jump from 3 to 6.75 suggests a potentially non-monotone behavior; depending on f and f', this could still converge to 5 in principle, but the numbers given do not strongly indicate a smooth approach to 5. Overall, this option portrays a pattern that does not clearly reflect stable convergence to 5.

Option B: "x2 = 6 and x3 cannot be calculated because f'(x2) = 0."
- This describes a breakdown of Newton's method due to a zero derivative at x2. Newton's method fails at the next step whenever f'(x2) = 0 because the update involves division by f'(x2). Such a scenario is entirely plausible for certain functions: if the tangent is horizontal at x2, the method halts. However, since we started at x1 = 1, this option asserts a specific obstacle occurring at x2. While possible, it is not something we can confirm from the information given about f or its derivatives. Therefore, while feasible, this option makes a particular structural claim about f that may or may not hold.

Option C: "The iterates x2, x3, x4, … oscillate and do not approach any zero."
- Newton's method can diverge or enter a cycle for some functions and initial guesses, especially if the function has particular pathologies or if the starting point lies in a region where the Newton map has a chaotic or non-convergent behavior. An oscillatory non-convergent behavior is a known possibility in root-finding with Newton’s method, particularly in complex scenarios or with poorly conditioned functions. However, with x1 = 1 and a target root at x = 5 (as implied by other options), this option asserts a non-convergent pattern without providing any structural reason (like f' vanishing or derivative blow-up) specific to f. It is a possible outcome in principle, but it would typically require additional information about f to justify.

Option D: "The iterates x2, x3, x4, … will approach the zero x = 5."
- This statement asserts convergence to the root at x = 5. Convergence in Newton's method depends on the function having a well-behaved derivative near the root and the starting value not being in a troublesome region. If the graph and the derivative indicate a simple root at x = 5 and the starting point x1 = 1 lies in a region where the Newton map guides iterates toward that root, then convergence to 5 is the expected behavior. Provided the graph shows a root at 5 and no derivative-zero barrier or chaotic dynamics near that path, this option aligns with the standard outcome of Newton’s method under favorable conditions. But without explicit details about f and f', we cannot confirm with certainty.

In summary, each option describes a possible Newton-step pattern under different scenarios:
- Option A gives a concrete sequence that does not obviously exhibit steady convergence to 5 and includes an intermediate overshoot.
- Option B posits a breakdown due to f'(x2) = 0, which is a valid failure mode of Newton's method but cannot be confirmed without f' known.
- Option C imagines non-convergence via oscillation, a legitimate possible behavior in some cases but requires more context.
- Option D asserts convergence to x = 5, which is the typical goal and plausible if the function and starting point are favorable.

Without the explicit function f and its derivatives, we cannot definitively adjudicate which pattern must occur. The right choice would be the one that correctly reflects the true Newton update given the specific f depicted in the provided graph. The other options describe alternative potential outcomes that could occur in other scenarios, but they would be incorrect for the actual function shown if the Newton iterations from x1 = 1 do indeed lead toward the root at x = 5 in a standard fashion.

MATH_1225_17255_202501 4.8 Newton's Method

The graph of 𝑦 = 𝑓 ( 𝑥 ) is given below. If Newton's method is applied to the function 𝑓 , starting with an initial guess of 𝑥 1 = 1 , what happens to the iterates 𝑥 2 , 𝑥 3 , 𝑥 4 , … ?

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