Calculate the first and second derivatives
The function is
f(x)=x^4-4x^3+20f(x)=x4−4x3+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)xcolor(white)(aaaa)-oocolor(white)(aaaa)0color(white)(aaaaaa)3color(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)uucolor(white)(aaaaaaa)nncolor(white)(aaaaaaaa)uu
graph{x^4-4x^3+20 [-32.73, 32.24, -5.85, 26.6]}