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

##### 1 Answer

Jan 22, 2016

Convex.

#### Explanation:

The sign of the second derivative is indicative of the function's convexity or concavity:

- If
#f''(-1)<0# , then#f(x)# is concave at#x=-1# . - If
#f''(-1)>0# , then#f(x)# is convex at#x=-1# .

Finding the second derivative requires a simple application of the power rule twice over:

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

#f'(x)=-27x^2-2x-3#

#f''(x)=-54x-2#

Find

#f''(-1)=-54(-1)-2=54-2=52#

Since **convex** at

graph{-3x^3-x^2-3x+2 [-2, 2, -10, 15]}