$(8$ marks$)$ Consider $f(x)=e^{2x}-4e^x.$

$($a$)$ Find the intervals of increase and decrease, and local extrema of $y=f(x).$

$[$Ans: Increasing on $(ln2,),$ Decreasing on $(-,ln2),$ Local min. $y=-4$

$($b$)$ Find the open intervals on which $f$ is concave up or concave down. Give all the inflection points, if there are any.

$(4$ marks$)$ Answer EXACTLY ONE of the following questions.

$($a$)$ The height of an arch is constructed above the ground is given by the function $h(x)=20cosx$ for $-\frac{}{2}x\frac{}{2}.$ What is the average height of the arch? $[$Ans: $\frac{40}{}]$

$($b$)$ Consider the function $f(x)=x^3-6x^2-4x+1.$ Starting with the initial point $x_0=0\mathrm{.}5,$ use the Newton's method to find the next point $x_1$ in the process of approximating the solution of the equation $f(x)=0.$

$[$Ans: $0\mathrm{.}243]$