# How do you determine whether the function f(x) = -x^4-9x^3+2x+4 is concave up or concave down and its intervals?

Nov 4, 2016

$f \left(x\right)$ is concave on $\left(- \infty , - 4.5\right)$ and $\left(0 , \infty\right)$, and $f \left(x\right)$ is convex on $\left(- 4.5 , 0\right)$.

#### Explanation:

To find where a function is concave up, find where the second derivative of the function is positive.

$f \left(x\right) = - {x}^{4} - 9 {x}^{3} + 2 x + 4$
Find $f ' \left(x\right)$:
$f ' \left(x\right) = - 4 {x}^{3} - 27 {x}^{2} + 2$
Next, find $f ' ' \left(x\right)$:
$f ' ' \left(x\right) = - 12 {x}^{2} - 54 x$
$f ' ' \left(x\right) = \left(- 6 x\right) \left(2 x + 9\right)$
Set $f ' ' \left(x\right)$ equal to zero to find inflection points
$0 = \left(- 6 x\right) \left(2 x + 9\right)$
$x = 0$, $x = - 4.5$

After checking the signs of values around these numbers, we find that $f ' ' \left(x\right)$ is positive on $\left(- 4.5 , 0\right)$ i.e. convex and $f ' ' \left(x\right)$ is negative on $\left(- \infty , - 4.5\right) \cup \left(0 , \infty\right)$ i.e. concave.
graph{-x^4-9x^3+2x+4 [-10, 5, -1000, 1000]}