Stationary points of a cubic

Statement

Given $f(x)=x^3-6x^2+9x+1$:

• (a) find the stationary points and classify them;
• (b) find the inflection point.

Solution

(a) $f'(x)=3x^2-12x+9=3(x-1)(x-3)$. Stationary points at $x=1$ and $x=3$. $f''(x)=6x-12$: $f''(1)=-6<0$ (maximum), $f''(3)=6>0$ (minimum). $f(1)=1-6+9+1=5$ (max), $f(3)=27-54+27+1=1$ (min).

(b) $f''(x)=0\Rightarrow x=2$, $f(2)=8-24+18+1=3$: inflection at $(2,3)$.

$\boxed{\text{max }(1,5);\ \text{min }(3,1);\ \text{inflection }(2,3)}$