# Is f(x)=-x^5+3x^4-9x^3-2x^2-6x concave or convex at x=8?

Apr 1, 2017

$f \left(x\right)$ is concave at $x = 8$. We know this by looking at the second derivative, which tells us about the concavity/shape of the graph.

#### Explanation:

When looking for the concavity of a function, it's best to find the second derivative, $f ' ' \left(x\right)$, of the function, $f \left(x\right)$.

When $f ' ' \left(x\right) < 0$, the $f \left(x\right)$ is concave
When $f ' ' \left(x\right) > 0$, the $f \left(x\right)$ is convex

The first derivative of this function is:
$f ' \left(x\right) = - 5 {x}^{4} + 12 {x}^{3} - 27 {x}^{2} - 4 x - 6$

The second derivative is:
$f ' ' \left(x\right) = - 20 {x}^{3} + 36 {x}^{2} - 54 x - 4$

Plug in $x = 8$ to get:
$f ' ' \left(8\right) = - 20 {\left(8\right)}^{3} + 36 {\left(8\right)}^{2} - 54 \left(8\right) - 4$
$f ' ' \left(8\right) = - 8372$

Since $f ' ' \left(8\right) < 0$, the $f \left(x\right)$ is concave at $x = 8$.