Question $532$ Marks $5\mathrm{.}1$ Using the fact that the derivative represents the slope $of$ the tangent line $to$ a graph, answer the following questions about $p(x)p(x)$ based $on$ its graph: $a.At$ which values $ofxxis$ the derivative $ofp(x)p(x)$ equal $to$ zero? $(2$ Marks$)b.$ Determine where $p(x)>0p(x)>0.(2$ Marks$)c.$ Determine where Marks$)d.$ Determine where $p(x)p(x)is$ undefined $(2$ Marks$)5\mathrm{.}2$ A cloud engineer $is$ analysing the average CPU usage per user for a machine learning service, where the function models the usage $,f(x)=ex\sqrt[\phantom{2}]{x}f(x)=$exx with representing the number $of$ simultaneous user requests. $To$ understand how the per$-$user CPU demand changes $as$ user load increases, the engineer needs $to$ determine the slope $of$ the tangent line $to$ the function $atx=4x=4.$ Calculate the slope $of$ the tangent line $tof(x)=ex\sqrt[\phantom{2}]{x}f(x)=$exx $at$ the point $x=4x=4.(6$ Marks$)5\mathrm{.}3$ Calculate the instantaneous rate $of$ change $atx=1x=1of$ the function, $f(x)=x--\sqrt[\phantom{2}]{ln(2x2-1)f(x)}=xln(2x2-1).$ Show all your calculations. $(8$ Marks$)5\mathrm{.}4$ Let $kkbe$ a continuous and differentiable function defined $on[0,2][0,2].$ Some function values $ofkk$ and $kk$ are given $in$ the table below: $xx0022323222k(x)k(x)33664400-2-2k(x)k(x)-3-33232-1-12211Ifp(x)=sin2(k(2x))p(x)=sin2(k(2x))$ evaluate $p(2)p(2).$ Explain your answer. $(10$ Marks$)$ End $of$ Question $5$

$5\mathrm{.}4$ Let $kbe$ a continuous and differentiable function defined $on0,2.$ Some function values $ofk$ and $k^{\text{'}}$ are given $in$ the table below:

$\begin{array}{cc}x& 0\\ \frac{}{2}& \\ \frac{3}{2}& 2\\ k(x)& 3\\ \frac{}{6}& \frac{}{4}\\ 0& -2\\ k^{\text{'}}(x)& -\frac{}{3}\\ \frac{3}{2}& -1\\ \frac{}{2}& 1\end{array}$

$Ifp(x)=si{n}^2(k(2x))$ evaluate $p^{\text{'}}(\frac{}{2}).$ Explain your answer.

$(10$ Marks$)$