# Is f(x)=4x^5-2x^4-9x^3-2x^2-6x concave or convex at x=-1?

Jan 22, 2016

Concave (also called "concave down").

#### Explanation:

Concavity and convexity are determined by the sign of the second derivative:

• If $f ' ' \left(- 1\right) < 0$, then the function is concave at $x = - 1$.
• If $f ' ' \left(- 1\right) > 0$, then the function is convex at $x = - 1$.

Find the second derivative:

$f \left(x\right) = 4 {x}^{5} - 2 {x}^{4} - 9 {x}^{3} - 2 {x}^{2} - 6 x$
$f ' \left(x\right) = 20 {x}^{4} - 8 {x}^{3} - 27 {x}^{2} - 4 x - 6$
$f ' ' \left(x\right) = 80 {x}^{3} - 24 {x}^{2} - 54 x - 4$

Find the sign of the second derivative when $x = - 1$:

$f ' ' \left(- 1\right) = 80 {\left(- 1\right)}^{3} - 24 {\left(- 1\right)}^{2} - 54 \left(- 1\right) - 4$

$= 80 \left(- 1\right) - 24 \left(1\right) + 54 - 4 = - 80 - 24 + 50 = - 54$

Since $f ' ' \left(- 1\right) < 0$, the function is concave at $x = - 1$. This means that it will resemble the $\cap$ shape. We can check a graph of $f \left(x\right)$:

graph{4x^5-2x^4-9x^3-2x^2-6x [-5, 5, -26.45, 19.8]}