Problem $12$

$-0\mathrm{.}25$

Let $f$ be a twice continuously differentiable function on $a,b$ such that $f^{\text{'}}(x)<mn>0</mn>$ and $f^{\text{'}\text{'}}(x)<mn>0</mn>$ for all xin$(a,b).$ Let $f()=0$ for some $in(a,b).$ The Newton's method to solve for $$ is given by

tilde$(x{)}_{k+1}=$tilde$(x{)}_k-\frac{f(tilde(x{)}_k)}{f^{\text{'}}(tilde(x{)}_k)}$

for some initial guess tilde$(x{)}_1.$ If tilde$(x{)}_{k}in(,b)$ for some $k0,$ then which of the following statement$($s$)$ is$/$are correct?

$($a$)$ tilde$(x{)}_{k+1}>$

$($b$)$ tilde

$($c$)$ tilde$(x{)}_{k+1}$

$(d)$ None $of$ the above