What are the values and types of the critical points, if any, of #f(x)=x^4 - 8x^3 + 18x^2 - 27#?

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Answer 1 · 2018-07-17T15:24:29

#x=3,f(3)=0# is an inflection point. and for #(0;-27)# a minimum point.

Your #f(x)# is equal to

#f(x)=(x+1)(x-3)^3#

so

#f'(x)=x(4x^2-24x+36)#
so

#x=0# or #x=3#

#f''(x)=12x^2-48x+36#

#f''(0)=36>0# we get a minimum

#f''(3)=0#

#f'''(x)=24x-48#

#f'''(3)=48ne 0#

since the derivative has an odd order we have an inflection point.

We have

#(0,-27)# a minimum

and for

#(3;0)# an inflection point.