We calculate the first derivative
#y=x^3-11x^2+39x-47#
#dy/dx=3x^2-22x+39#
We need #dy/dx=0#
#3x^2-22x+39=(3x-13)(x-3)=0#
The critical points are
#x=13/3# and #x=3#
Now, we construct the chart
#color(white)(aa)##Interval##color(white)(aa)##|##color(white)(aaa)##]-oo,3[##color(white)(aaa)##|##color(white)(aaa)##]3,13/3[##color(white)(aaa)##|##]13/3,+oo[#
#color(white)(aa)##x-3##color(white)(aaaaa)##|##color(white)(aaa)##-##color(white)(aaaaaaa)##|##color(white)(aaaaaa)##+##color(white)(aaaa)##|##color(white)(aaaa)##+#
#color(white)(aa)##3x-13##color(white)(aaa)##|##color(white)(aaa)##-##color(white)(aaaaaaa)##|##color(white)(aaaaaa)##-##color(white)(aaaa)##|##color(white)(aaaa)##+#
#color(white)(aa)##dy/dx##color(white)(aaaaaaa)##|##color(white)(aaa)##+##color(white)(aaaaaaa)##|##color(white)(aaaaaa)##-##color(white)(aaaa)##|##color(white)(aaaa)##+#
#color(white)(aa)##y##color(white)(aaaaaaaaa)##|##color(white)(aaa)##↗##color(white)(aaaaaa)##|##color(white)(aaaaaa)##↘##color(white)(aaaa)##|##color(white)(aaaa)##↗#
The intervals of increasing are #]-oo,3[# and #]13/3,+oo[#
The interval of decreasing is #]3,13/3[#
