The graph of the function $f$ is shown below. Let g be the function given by $g(x)={\displaystyle \underset{x}{\overset{-4}{}}}f(t)dt.$

a$.$ Find all the values of x in the closed interval $-8x8$ for which $g(x)=0.$

b$.$ Find the average rate of change of g on the interval $-4x6.$

c$.$ For how many values of $c,$ where $-4c6,$ is $g^{\text{'}}(c)$ equal to the average rate found in part $($b$)?$ Explain your reasoning.

b$.$ Find the $x-$coordinate of each local maximum of $g$ on the open interval $fhh(x)={\displaystyle \underset{-4}{\overset{x}{}}}f(t)dth(0),h^{\text{'}}(0)h^{\text{'}\text{'}}(0)t=-4t=4t=-4t=4hh-4,4-8.$ Justify your answer.

The graph $offis$ shown below. Let $hbe$ the function given $byh(x)={\displaystyle \underset{-4}{\overset{x}{}}}f(t)dt.$

$a.$ For each $ofh(0),h^{\text{'}}(0),$ and $h^{\text{'}\text{'}}(0),$ find the value $or$ explain why $it$ does not exist.

$b.$ Find the total distance traveled $by$ the particle

$c.$ Find the total displacement $of$ the particle from from $t=-4tot=4.t=-4tot=4.$

$d.$ Find the relative extrema and inflection points $ofhto$ help sketch the graph $ofhon-4,4.$