Let $f:[a, b] \rightarrow \mathbb{R}$ (the set of all real numbers) be any function which is twice differentiable in $(a, b)$ with only one root $\alpha$ in $(a, b)$. Let $f^{\prime}(x)$ and $f^{\prime \prime}(x)$ denote the first and second order derivatives of $f(x)$ with respect to $x .$ If $\alpha$ is a simple root and is computed by the Newton-Raphson method, then the method converges if
(A) $\left|f(x) f^{\prime \prime}(x)\right|<\left|f^{\prime}(x)\right|^{2}$, for all $x \in(a, b)$
(B) $\left|f(x) f^{\prime}(x)\right|<\left|f^{\prime \prime}(x)\right|$, for all $x \in(a, b)$
(C) $\left|f^{\prime}(x) f^{\prime \prime}(x)\right|<|f(x)|^{2}$, for all $x \in(a, b)$
(D) $\left|f(x) f^{\prime \prime}(x)\right|<\left|f^{\prime}(x)\right|$, for all $x \in(a, b)$

Question ID - 167050 | SaraNextGen Top Answer

Let $f:[a, b] \rightarrow \mathbb{R}$ (the set of all real numbers) be any function which is twice differentiable in $(a, b)$ with only one root $\alpha$ in $(a, b)$. Let $f^{\prime}(x)$ and $f^{\prime \prime}(x)$ denote the first and second order derivatives of $f(x)$ with respect to $x .$ If $\alpha$ is a simple root and is computed by the Newton-Raphson method, then the method converges if
(A) $\left|f(x) f^{\prime \prime}(x)\right|<\left|f^{\prime}(x)\right|^{2}$, for all $x \in(a, b)$
(B) $\left|f(x) f^{\prime}(x)\right|<\left|f^{\prime \prime}(x)\right|$, for all $x \in(a, b)$
(C) $\left|f^{\prime}(x) f^{\prime \prime}(x)\right|<|f(x)|^{2}$, for all $x \in(a, b)$
(D) $\left|f(x) f^{\prime \prime}(x)\right|<\left|f^{\prime}(x)\right|$, for all $x \in(a, b)$