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

We are given the fixed-point iteration schemes for solving the equation \( x^3 + 4x^2 - 10 = 0 \) in the interval \( \left( 1, \frac{3}{2} \right) \).

Step 1: Analyzing Statement S1:
\[ x_{k+1} = \sqrt{\frac{10}{4 + x_k}}, \quad k = 0, 1, 2, \ldots \]
This is a valid fixed-point iteration scheme. To determine convergence, we examine the derivative of the function \( g(x) = \sqrt{\frac{10}{4 + x}} \) at the root. For convergence, we need:
\[ |g'(x)| < 1 \]
For \( g(x) \), the derivative is:
\[ g'(x) = -\frac{10}{2(4 + x)^{3/2}} \]
Evaluating this at \( x = 1 \) gives a value less than 1, confirming that this iteration scheme converges for any initial guess in the interval \( \left( 1, \frac{3}{2} \right) \). Hence, S1 is TRUE.

Step 2: Analyzing Statement S2:
\[ x_{k+1} = \frac{1}{2} \sqrt{10 - x_k^3}, \quad k = 0, 1, 2, \ldots \]
This iteration scheme may diverge for some initial guesses in the given interval. To check for convergence, we again compute the derivative of the iteration function:
\[ g'(x) = \frac{3x^2}{2 \sqrt{10 - x^3}} \]
At \( x = 1 \), we find that \( |g'(1)| > 1 \), indicating that the scheme may diverge for some initial guesses. Therefore, S2 is FALSE.

Thus, the correct answer is Option A.

\[ \boxed{A} \quad \text{S1 is TRUE and S2 is FALSE} \]