Answer
$ a.\quad$ g is odd
$ b.\quad$local maximum of $54$ at $x=-3$.
Work Step by Step
$a.$
$g(-x)=(-x)^{3}-27(-x)=-x^{3}+27x=-(x^{3}-27x)=-g(x)$
$g(-x)=-g(x)$, so the function is odd
$b.$
Since the function is odd, the graph has origin symmetry.
If $(3,-54)$ is a local minimum point, then the point symmetric to it,
$(-3,54)$ is on the graph and is a local maximum point.
