QUESTIGN ONE $8$Pts$.$

a$]$ If the function $f(x)$ is continuous in a closed interval $a,b$ and $f(a)f(b)>0,$ does $f(x)$ has a root in $a,b?,$ Justify and explain geometrically,

b$]$ Derive the sufficient condition for the convergence of Newton's Method,

c$]$ Explain geometrically one example where Newton's Method fail to converge.

d$]$ Use the fixed$-$point iteration method to find the root of $2x=cosx+3$ correct to three decimal places, check convergence of the method geometrically and analytically before calculations

Hint let $x_0=\frac{}{2}$