# Is f(x)=-x^5-5x^3-3x^2+12x+8 concave or convex at x=-3?

Mar 6, 2017

The curve is convex at $x = - 3$

#### Explanation:

We calculate the first and second derivative

$f \left(x\right) = - {x}^{5} - 5 {x}^{3} - 3 {x}^{2} + 12 x + 8$

$f ' \left(x\right) = - 5 {x}^{4} - 15 {x}^{2} - 6 x + 12$

$f ' ' \left(x\right) = - 20 {x}^{3} - 30 x - 6$

And

$f \left(- 3\right) = - 20 \cdot {\left(- 3\right)}^{3} - 30 \cdot \left(- 3\right) - 6$

$= - 20 \cdot \left(- 27\right) + 90 - 6$

$= 624$

Therefore,

As $f ' ' \left(- 3\right) > 0$, the curve is convex at $x = - 3$