How do you find all critical point and determine the min, max and inflection given f(x)=x^4-4x^3+20f(x)=x44x3+20?

1 Answer
Jul 6, 2018

Please see the explanation below

Explanation:

Calculate the first and second derivatives

The function is

f(x)=x^4-4x^3+20f(x)=x44x3+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]}