Calculate the first and second derivatives

The function is

#f(x)=x^4-4x^3+20#

Calculate the first derivative

#f'(x)=4x^3-12x^2#

#f'(x)=0#

#=>#, #4x^3-12x^2=0#

#=>#, #4x^2(x-3)=0#

The critical points are #x=0# and #x=3#

Construct a variation chart

#color(white)(aaaa)##x##color(white)(aaaa)##-oo##color(white)(aaaa)##0##color(white)(aaaaaa)##3##color(white)(aaaa)##+oo#

#color(white)(aaaa)##f'(x)##color(white)(aaaaa)##-##color(white)(aaaa)##-##color(white)(aaaa)##+#

#color(white)(aaaa)##f(x)##color(white)(aaaaa)##↘##color(white)(aaaa)##↘##color(white)(aaaa)##↗#

There is a local minimum at #(3, -7)#

Calculate the second derivative

#f''(x)=12x^2-24x#

The points of inflections are when #f''(x)=0#

#12x^2-24x=0#

#=>#, #12x(x-2)=0#

#=>#, #x=0# and #x=2#

The inflection points are #(0, 20)# and #(2,4)#

Build a variation chart to determine the concavities

#color(white)(aaaa)##" Interval "##color(white)(aaaa)##(-oo, 0)##color(white)(aaaa)##(0,2)##color(white)(aaaa)##(2,+oo)#

#color(white)(aaaa)##" sign f''(x)"##color(white)(aaaaaaa)##+##color(white)(aaaaaaa)##-##color(white)(aaaaaaaa)##+#

#color(white)(aaaa)##" f(x)"##color(white)(aaaaaaaaaaaa)##uu##color(white)(aaaaaaa)##nn##color(white)(aaaaaaaa)##uu#

graph{x^4-4x^3+20 [-32.73, 32.24, -5.85, 26.6]}