$[9$pts$]$ Let $f(t)$ be the function in the graph below and let $g(x)={\displaystyle \underset{-5}{\overset{x}{}}}f(t)dt.$

$($a$)$ Evaluate

$g(-3)=$

$g(-1)=$

$g(5)-g(1)=$

$($b$)$ Compute $g^{\text{'}}(x).$ Justify your answer.

$($c$)$ Find the $x$ values where $g$ attains a local maximum or minimum. Explain how you know.

$($d$)$ Make a rough ${?}^1$ sketch of $g(x)$ below.

${?}^1$ It's okay if the $y$ values and slopes of tangent lines aren't perfectly accurate, but try to get at least some correct