Answer
a. f is odd
b. local minimum of $-16$ at $x=−2$
Work Step by Step
a.
$f(−x)=-(−x)^3+12(−x)= x^3-12x=−(-x^3+12x)=−f(x)$
$f(−x)=−f(x)$, so the function is odd
b.
Since the function is odd, the graph has origin symmetry.
If (2,16) is a local maximum point, then the point symmetric to it,
(−2,-16) is on the graph and is a local minimum point.