How do you find the inflection points for the given function f(x) = x^3 - 3x^2 + 6x?

Jul 24, 2018

Below

Explanation:

$f \left(x\right) = {x}^{3} - 3 {x}^{2} + 6 x$
$f ' \left(x\right) = 3 {x}^{2} - 6 x + 6$
$f ' ' \left(x\right) = 6 x - 6$

Inflexion points can be found when the second derivative is equal to 0
For inflexion points, $f ' ' \left(x\right) = 0$
$6 x - 6 = 0$
$x = 1$

Test $x = 1$ for concavity
When $x = 0$, $f ' ' \left(0\right) = - 6$
When $x = 1$, $f ' ' \left(1\right) = 0$
When $x = 2$, $f ' ' \left(2\right) = 6$
Therefore, since there is a change in concavity, there is a point of inflexion at $\left(1 , 4\right)$