# How do you find all points of inflection of the function f(x)=x^3-3x^2-x+7?

May 20, 2015

The inflection of a function means the changes in its direction. You **can either understand it as the change from increasing to decreasing (and vice-versa) on your $f \left(x\right)$ or the vertex of the slope for your $f ' \left(x\right)$.

But how do we find inflection points? Using your function's second derivatives $f ' ' \left(x\right)$.

Just a quick reminder: your $f ' \left(x\right)$ measures the increase / decrease rate of $f \left(x\right)$, while $f ' ' \left(x\right)$ indicates whether $f \left(x\right)$ changes from convex to concave (or vice-versa).

Thus, we must equal the second derivative of the function to zero.

$f ' \left(x\right) = 3 {x}^{2} - 6 x - 1$,
$f ' ' \left(x\right) = 6 x - 6$.

Equaling it to zero, we have $6 x = 6$, $x = 1$.

For $f \left(1\right) = {1}^{3} - 3 \left({1}^{2}\right) - 1 + 7 = 4$

Thus, your inflection point is $\left(1 , 4\right)$

graph{x^3-3x^2-x+7 [-15.27, 16.33, -2.72, 13.08]}