# How do you find the inflection points of the graph of the function f(x) = x^3 - 3x^2 + 3x?

Apr 6, 2015

Find the points on the graph where the concavity changes.

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

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

And $f ' ' \left(x\right) = 6 x - 6$.

$f ' ' \left(x\right) = 0$ for $x = 1$. Testing on each side of $1$ we find that

$f ' ' \left(x\right) < 0$ (so the graph of $f$ is concave down) for $x < 1$
$f ' ' \left(x\right) > 0$ (so the graph of $f$ is concave up) for $x > 1$

At $x = 1$, we have $y = f \left(1\right) = 3 - 3 + 1 = 1$.

The inflection point is $\left(1 , 1\right)$.

Apr 6, 2015

You must study your second derivative: