Given f$($x$)=$ x

$34$x $+2,$ solve the following problems.

$($a$)$ Verify that f$($x$)=$ x

$34$x $+2$ has a root in $[0\mathrm{.}4,0\mathrm{.}7].$

$($b$)$ Use the bisection method to find p$3$ for f$($x$)=$ x

$34$x $+2$ on $[0\mathrm{.}4,0\mathrm{.}7]$ by hand

calculation $($i$.$e$.,$ do not use code and do not check stopping criteria$).$ Do

your work with at least $6$ decimal digits if a number has more than $6$ digits.

$($c$)$ Apply the bisection method to find approximate root of f$($x$)=$ x

$34$x $+2$ with

$=106$

in $[0\mathrm{.}4,0\mathrm{.}7]$ by using the code $$alg$021$Bisection.m$.$ Turn in a copy of

the $$command window$$ including all input and output.

$($d$)$ Find a bound for the number of iterations needed to achieve an approximation

with accuracy $=106$

to the root of f$($x$)=$ x

$34$x $+2$ in $[1\mathrm{.}2,1\mathrm{.}4].($Use the

result obtained in Theorem $2\mathrm{.}1\mathrm{.}3$ on p$.29$ in lecture notes or Theorem $1$ on p$.18$

in slides of Ch$.2.)$ Is such bound consistent with the number of iterations needed

when executing the code done in part $($c$)?$