We calculate the first derivative and construct a sign chart.
We need
(u/v)'=(u'v-uv')/(v^2)
f(x)=x^3/(x^2-4)
u=x^3, =>, u'=3x^2
v=x^2-4, =>, v'=2x
f'(x)=(3x^2(x^2-4)-2x*x^3)/(x^2-4)^2
=(3x^4-12x^2-2x^4)/(x^2-4)^2
=(x^4-12x^2)/(x^2-4)^2
=(x^2(x^2-12))/(x^2-4)^2
f'(x)=(x^2(x+sqrt12)(x-sqrt12))/((x+2)^2(x-2)^2)
We construct the sign chart
color(white)(aaaa)xcolor(white)(aaaaaaa)-oocolor(white)(aaaa)-sqrt12color(white)(aaa)-2color(white)(aaa)0color(white)(aaa)2color(white)(aaa)sqrt12color(white)(aaa)+oo
color(white)(aaaa)x+sqrt12color(white)(aaaaaaaa)-color(white)(aaaa)+color(white)(aaa)+color(white)(a)+color(white)(aa)+color(white)(aaaa)+
color(white)(aaaa)x+2color(white)(aaaaaaaaaaa)+color(white)(aaaa)+color(white)(aaa)+color(white)(a)+color(white)(aa)+color(white)(aaaa)+
color(white)(aaaa)x-2color(white)(aaaaaaaaaaa)+color(white)(aaaa)+color(white)(aaa)+color(white)(a)+color(white)(aa)+color(white)(aaaa)+
color(white)(aaaa)x-sqrt12color(white)(aaaaaaaa)-color(white)(aaaa)-color(white)(aaa)-color(white)(a)-color(white)(aa)-color(white)(aaaa)+
color(white)(aaaa)f'(x)color(white)(aaaaaaaaaaa)+color(white)(aaaa)-color(white)(aaa)-color(white)(a)-color(white)(aa)-color(white)(aaaa)+
color(white)(aaaa)f(x)color(white)(aaaaaaaaaa)↗color(white)(aaaa)↘color(white)(aaa)↘color(white)(a)↘color(white)(aa)↘color(white)(aaaa)↗
The relative maximum is when x=-sqrt12
The relative minimum is when x= sqrt12
graph{x^3/(x^2-4) [-14.24, 14.24, -7.12, 7.12]}