Question # $1$

The function f$($x$)=$ e

$$x $$ cos$($x$),$ with x in $[0,\backslash$infty $).$ Answer the following questions,

and show all details of your work.

$(1)$ Determine the approximate location of the roots graphically. Validate analytically the graphical result for one of them, in case there are multiple

roots$/$zeros$.($Analytical verification implies obtaining a power series polynomial expansion of f$($x$)$ and solving that approximately.$)(20$ points$)$

$(2)$ For the approximate root you found in part $(1),$ find its improved version via

the bisection method for tolerance values $1\backslash$times $1005,1\backslash$times $1008$ & $1\backslash$times $1010$

$.$

How many iterations were required in each case? $(20$ points$)$

$(3)$ For the approximate root you found in part $(1),$ find its improved version via

the Newton$-$Raphson method for tolerance values $1\backslash$times $1005,1\backslash$times $1008$ & $1\backslash$times

$1010.$ How many iterations were required in each case? What is a restriction that is very specific when applying the Newton$-$Raphson method? $(25$

points$)$

$(4)$ Using the final values for the roots you obtained in part $(2)$ as initial guess

for Newton$-$Raphson method, find the final values for the tolerance values