Question

To find a real root of the equation \( x^3 + 4x^2 - 10 = 0 \) in the interval \( \left( 1, \frac{3}{2} \right) \) using the fixed-point iteration scheme, consider the following two statements:
Statement 1 S1: The iteration scheme \( x_{k+1} = \sqrt{\frac{10}{4 + x_k}}, \, k = 0, 1, 2, \ldots \) converges for any initial guess \( x_0 \in \left( 1, \frac{3}{2} \right) \). 
Statement 2 S2: The iteration scheme \( x_{k+1} = \frac{1}{2} \sqrt{10 - x_k^3}, \, k = 0, 1, 2, \ldots \) diverges for some initial guess \( x_0 \in \left( 1, \frac{3}{2} \right) \).

• A. \( \text{S1 is TRUE and S2 is FALSE} \)
• B. \( \text{S2 is TRUE and S1 is FALSE} \)
• C. \( \text{Both S1 and S2 are TRUE} \)
• D. \( \text{Neither S1 nor S2 is TRUE} \)

Hint:

For fixed-point iteration schemes, check the convergence by evaluating the derivative of the iteration function. If \( |g'(x)|<1 \), the scheme converges; if \( |g'(x)|>1 \), the scheme diverges.

Answer 1

The Correct Option is A

To determine the validity of the statements regarding the fixed-point iteration schemes, we need to analyze each scheme's convergence in the given interval \( (1, \frac{3}{2}) \).

Analysis of Statement 1 (S1)

The iteration scheme given is: \(x_{k+1} = \sqrt{\frac{10}{4 + x_k}}\). 

To check the convergence of this iteration method, we need to determine if the derivative of the function inside the iteration, \( f(x) = \sqrt{\frac{10}{4 + x}} \), has an absolute value less than 1 within the interval.

First, let's differentiate \( f(x) \):

\(f(x) = \sqrt{\frac{10}{4 + x}} \implies f'(x) = \frac{-5}{(4 + x)^{3/2}}\)

For convergence, we require:

\(|f'(x)| < 1\)

Evaluate \( f'(x) \) within the interval \( (1, \frac{3}{2}) \):

• At \( x = 1 \): \(f'(1) = -\frac{5}{(5)^{3/2}} = -\frac{5}{5\sqrt{5}} = -\frac{1}{\sqrt{5}} \approx -0.447\)
• At \( x = \frac{3}{2} \): \(f'(\frac{3}{2}) = -\frac{5}{(5.5)^{3/2}} \approx -0.404\)

Since \(|f'(x)| < 1\) for \( x \in (1, \frac{3}{2}) \), the iteration scheme converges for all initial guesses in this interval.

Conclusion for S1: TRUE

Analysis of Statement 2 (S2)

The second iteration scheme is: \(x_{k+1} = \frac{1}{2} \sqrt{10 - x_k^3}\).

We analyze the convergence by finding the derivative of \( g(x) = \frac{1}{2} \sqrt{10 - x^3} \).

Differentiate \( g(x) \):

\(g'(x) = \frac{1}{2} \cdot \frac{-3x^2}{2\sqrt{10 - x^3}} = -\frac{3x^2}{4\sqrt{10 - x^3}}\)

For convergence, we require:

\(|g'(x)| < 1\)

Check \( g'(x) \) within the interval \( (1, \frac{3}{2}) \):

• At \( x = 1 \): \(g'(1) = -\frac{3}{4\sqrt{9}} = -\frac{1}{4} \approx -0.25\)
• At \( x = \frac{3}{2} \): \(g'(\frac{3}{2}) = -\frac{3 \times (\frac{9}{4})}{4\sqrt{10 - \frac{27}{8}}} = -\frac{27}{16\sqrt{\frac{53}{8}}} \approx -1.02\)

Since \(|g'(x)| > 1\) for some \( x \) in \( (1, \frac{3}{2}) \), the iteration may diverge.

Conclusion for S2: TRUE

Final Answer

The correct interpretation and solving steps reveal that the accurate result is: S1 is TRUE and S2 is TRUE. Therefore, there seems to be an inconsistency in the given correct answer in context with our analysis.