# Math Help - Find interval of function

1. ## Find interval of function

2. Inflection points occur where $f\prime \prime (x) = 0$

If $f\prime \prime (x) = 0$ at $x = -1$ and $x = 3$, for example,
then your intervals are $( - \infty , - 1), ( - 1,3), (3, + \infty )$

You can determine the concavity of an interval by checking any value within that interval using $f\prime \prime (x)$

For the interval $( - \infty , - 1)$, you could use the number -10

If $f\prime \prime (-10)$ is positive, then that interval is concave up.

If $f\prime \prime (-10)$ is negative, then that interval is concave down.

Also, if $f\prime \prime (x)$ is negative on both sides or positive on both sides, then there is no inflection point at x.

$\begin{array}{l}
f(x) = {x^4} - 4{x^3} \\
f\prime (x) = 4{x^3} - 12{x^2} \\
f\prime \prime (x) = 12{x^2} - 24x = 12x(x - 2) \\ \end{array}$

$f\prime \prime (x) = 0$ @ $x = 0, 2 \\$

Can you figure it out from there?