# Is #f(x)=-2x^3-2x^2+8x-1# concave or convex at #x=3#?

##### 1 Answer

Feb 8, 2016

Concave (sometimes called "concave down")

#### Explanation:

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

- If
#f''(3)<0# , then#f(x)# is concave at#x=3# . - If
#f''(3)>0# , then#f(x)# is convex at#x=3# .

To find the function's second derivative, use the power rule repeatedly.

#f(x)=-2x^3-2x^2+8x-1#

#f'(x)=-6x^2-4x+8#

#f''(x)=-12x-4#

The value of the second derivative at

#f''(3)=-12(3)-4=-40#

Since this is

These are the general shapes of concavity (and convexity):

We can check the graph of the original function at

graph{-2x^3-2x^2+8x-1 [-4,4, -150, 40]}