How do you find all critical point and determine the min, max and inflection given f(x)=x^3-6x^2+9x+8?

1 Answer
Apr 19, 2018

See below

Explanation:

Given f(x), his critical point are given by f´(x)=0. Lets calculate

f´(x)=3x^2-12x+9=0

Using quadratic formula x=(12+-sqrt(144-108))/6=(12+-6)/6

One critical point is x=3 and the other is x=1

Analyzing the sign of f´(x) in intervals (-oo,1) (1,3) and (3,+oo) we found

f´(x)>0 in (-oo,1) so f is increasing there
f´(x)<0 in (1,3) so f is decreasing there
f´(x)>0 in (3,+oo) so f is increasing there

Sumarizing f(x) has a maximum in x=1, has a minimum in x=3 graph{x^3-6x^2+9x+8 [-15.25, 25.33, -3, 17.27]}

If we calculate f´´(x)=6x-12=0 we found x=2 as infexion point