Which of the following statements are true and which are false?

Question

Which of the following statements are true and which are false? Give a short proof or a counterexample in support of your answer:

i) The equation x³ 4x 16 = 0 has a root in the interval [3, 4].

ii) The order of convergence of the secant method is 0.62.

iii) For the system of linear equations:

5x + y + 2z = 34

4y − 3z = 12

10x − 2y + z = −4

the matrix is diagonally dominant.

iv) The numerical method:

$y_{n+1}=\left(1+\lambda h+\frac{\lambda^2 h^2}{2}\right)y_n,\quad\lambda>0,\quad n\geq 0$

is relatively stable.

v) The method:

$x_{n+1}=\frac{x_n}{2}+\frac{9}{8x_n}$

converges to 1.5 for any choice of initial approximation.

12 Sep 2025

Answer :

Word Count : 255

i) True. Consider the function $f(x) = x^3 - 4x - 16$. Evaluating at the endpoints of the interval: $f(3) = 27 - 12 - 16 = -1$ and $f(4) = 64 - 16 - 16 = 32$. Since $f(3) < 0$ and $f(4) > 0$, by the Intermediate Value Theorem, ______ ___ _________ _________ ___.
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