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

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 : 433

i) The equation $x^3 - 4x + 16 = 0$ has a root in the interval \[3, 4]. To check, we can use the Intermediate Value Theorem (IVT). Define $f(x) = x^3 - 4x + 16$. Compute $f(3)$ and $f(4)$: $$ f(3) = 3^3 - 4 \cdot 3 + 16 = 27 - 12 + 16 = 31 $$ $$ f(4) = 4^3 - 4 \cdot 4 + 16 = 64 - 16 + 16 = 64 $$ Both $f(3)$ and $f(4)$ are positive, so $f(x)$ ___ _____ ____ __________ ________ ___ _________ ______.
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