$(22$ points$)$ Let $g(x)=\frac{x^3-3x^2-9x-1,}{3x^2-6x-9x-1},g^{\text{'}}(x)=3x^2-6x-9,g^{\text{'}\text{'}}(x)=6x-6$

$($a$)(3$ points$)$ Find the interval $($s$)$ on which $g(x)$ is concave up$.(-,)(,)$axis

$g^{\text{'}\text{'}}(-1)=6(-1)-6=-12$ concave down

$g^{\text{'}\text{'}}(1)=6(1)-6=$ concave up

$($b$)(3$ points$)$ Find the interval$($s$)$ on which $g(x)$ is concave down.$)$

$($c$)(4$ points$)$ Find the inflection point$($s$)$ of $g(x).$

$g(x)=(1{)}^3-3(1{)}^2-g(1)-1$

$g(x)=-12$

$+6=6$

$+6x=6$

$x=6$

Q$)6$ INFLECTION POINT

$x=1$ I $x=(1,-12)$

$($d$)(6$ points$)$ Find all the local minhtrum and maximum values of $g(x).,x=(1,-12)$

$($e$)(6$ points$)$ Find the global maximum and minimum values of $g(x)$ on the interval $-2,10.$