Let $g(x{)}_{?}=\frac{f(t)dti,0}{?}x7$ where $f$ is pictured.

$($a$)$ At what value of $x$ does $g(x)$ achieve local maximum? Explain your reasoning.

$($b$)$ Over what subintervals of $0,7$ is $g(x)$ increasing? Explain your reasoning.

$($c$)$ Over what subintervals of $0,7$ is $g(x)$ concave up$?$ Explain your reasoning.

$($d$)$ Find $g(3).$ Explain your reasoning and show any calculations necessary to answer this part.