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

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Answer 1 · 2018-04-19T12:46:21

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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

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