Study of y = x³ − 6x² + 9x

Text

Study the function $y=x^{3}-6x^{2}+9x$: zeros and sign, relative maxima and minima, inflection point.

Solution

Zeros and sign. $y=x(x-3)^{2}$: zeros at $x=0$ and $x=3$ (double). For $x<0$, $y<0$; for $x>0$ ($x\neq3$), $y>0$, tangent to the axis at $x=3$. Extrema. $y'=3x^{2}-12x+9=3(x-1)(x-3)$, vanishing at $x=1,3$. With $y''=6x-12$: at $x=1$, $y''<0$ maximum, $y(1)=4$; at $x=3$, $y''>0$ minimum, $y(3)=0$. Inflection. $y''=0\Rightarrow x=2$, with $y(2)=2$. $\boxed{\text{max }(1,4);\quad \text{min }(3,0);\quad \text{inflection }(2,2)}$