We start with the first derivative
#y=x^3-6x^2-36x+16#
#dy/dx=3x^2-12x-36#
The critical points are when #dy/dx=0#
That is,
#3x^2-12x-36=0#
#2(x^2-4x-12)=0#
#2(x+2)(x-6)=0#
Therefore,
#x=-2# and #x=6#
We build a sign chart
#color(white)(aaaa)##x##color(white)(aaaa)##-oo##color(white)(aaaa)##-2##color(white)(aaaa)##6##color(white)(aaaa)##+oo#
#color(white)(aaaa)##x+2##color(white)(aaaaa)##-##color(white)(aaaa)##+##color(white)(aaaa)##+#
#color(white)(aaaa)##x-6##color(white)(aaaaa)##-##color(white)(aaaa)##-##color(white)(aaaa)##+#
#color(white)(aaaa)##dy/dx##color(white)(aaaaaaa)##+##color(white)(aaaa)##-##color(white)(aaaa)##+#
#color(white)(aaaa)##y##color(white)(aaaaaaaaa)##↗##color(white)(aaaa)##↘##color(white)(aaaa)##↗#
Now, we calculate the second derivative
#(d^2y)/dx^2=6x-12#
We have an inflexion point when, #(d^2y)/dx^2=0#
That is, #x=2#
We make a second chart
#color(white)(aaaa)##Interval##color(white)(aaaa)##]-oo,2[##color(white)(aaaa)##]2,+oo[#
#color(white)(aaaa)##(d^2y)/dx^2##color(white)(aaaaaaaaaa)##-##color(white)(aaaaaaaa)##+#
#color(white)(aaaa)##y##color(white)(aaaaaaaaaaaaa)##nn##color(white)(aaaaaaaa)##uu#
We have a local maximum at #(-2,56)# and a local minimum at #(6,-200)# and an inflexion point at #(2,-72)#
