# Is f(x)=-x^3+2x^2-2x-2 concave or convex at x=1?

May 24, 2016

concave at x = 1

#### Explanation:

To determine if f(x) is concave/convex at x = a , we consider the value of f''(a).

• If f''(a) > 0 , then f(x) is convex at x = a

• If f'')a) < 0 , then f(x) is concave at x = a

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

differentiate using the $\textcolor{b l u e}{\text{power rule}}$

$\Rightarrow f ' \left(x\right) = - 3 {x}^{2} + 4 x - 2$

and $f ' ' \left(x\right) = - 6 x + 4$

$\Rightarrow f ' ' \left(1\right) = - 6 + 4 = - 2$

Since f''(1) < 0 , then f(x) is concave at x = 1
graph{-x^3+2x^2-2x-2 [-10, 10, -5, 5]}