First calculate, the first derivative
#y=x^4-8x^3#
#y'=4x^3-24x^2=4x^2(x-6)#
The critical points are when #y'=0#
#4x^2(x-6)=0#
#=>#, #{(x=0),(x=6):}#
Build a variation chart
#color(white)(aaaa)##x##color(white)(aaaa)##-oo##color(white)(aaaa)##0##color(white)(aaaaaa)##6##color(white)(aaaaa)##+oo#
#color(white)(aaaa)##x^2##color(white)(aaaaaaaa)##+##color(white)(aaaa)##+##color(white)(aaaaa)##+#
#color(white)(aaaa)##x-6##color(white)(aaaaaa)##-##color(white)(aaaa)##-##color(white)(aaaaa)##+#
#color(white)(aaaa)##y'##color(white)(aaaaaaaaa)##-##color(white)(aaaa)##-##color(white)(aaaaa)##+#
#color(white)(aaaa)##y##color(white)(aaaaaaaaa)##↘##color(white)(aaaa)##↘##color(white)(aaaaa)##↗#
There is a local minimum at #(6, -432)#
Calculate the second derivative
#y''=12x^2-48x=12x(x-4)#
The inflection points are when #y''=0#
#12x(x-4)=0#
#=>#, #{(x=0),(x=4):}#
Build a variation chart
#color(white)(aaaa)##" Interval "##color(white)(aaaa)##(-oo,0)##color(white)(aaaa)##(0,4)##color(white)(aaaaaa)##(4,+oo)#
#color(white)(aaaaaa)##"Sign y'' "##color(white)(aaaaaaa)##+##color(white)(aaaaaaa)##-##color(white)(aaaaaaaaa)##+#
#color(white)(aaaaaa)##" y "##color(white)(aaaaaaaaaaa)##uu##color(white)(aaaaaaa)##nn##color(white)(aaaaaaaaa)##uu#
The inflection points are #(0,0)# and #(4,-256)#
The curve is convex when # x in (-oo,0) uu(4, +oo)#
The curve is concave when # x in (0,4)#
graph{x^4-8x^3 [-25.66, 25.67, -12.83, 12.83]}