The continuous function f is defined on the closed interval $-4x4.$ The figure above shows a portion of the graph of $f,$ consisting of two line segments and a graph of a quadratic function with an opening down. The function $g$ is given by $g(x)={\displaystyle \underset{-4}{\overset{x}{}}}f(t)dt.$

$($a$)$ find $g(1).$

$($b$)$ On what open intervals contained in $ghh(x)=\frac{g(x)}{x}{h}^{\text{'}}(-1)g(x)=0x-4x-4is$ the graph $ofg$ both increasing and concave down? Give a reason for your answer.

$(c)$ The function $his$ defined $byh(x)=\frac{g(x)}{x},$ find $h^{\text{'}}(-1).$

$(d)ifg(x)=0,$ How many possible $x$ values are there $on$ the closed interval $-4x4?$ Justify this result.