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

1 Answer
Jan 28, 2016

Concave (this is also called concave down).

Explanation:

The concavity or convexity of a function are determined by the sign of the second derivative.

  • 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 of the function is a simple application of the power rule.

#f(x)=9x^3+2x^2-2x-2#

#f'(x)=27x^2+4x-2#

#f''(x)=54x+4#

Find the sign of the second derivative at #x=-1#:

#f''(-1)=-54+4=-50#

Since this is #<0#, the function is concave at #x=-1#. Concavity means that the function resembles the #nn# shape. We can check the graph of #f(x)#:

graph{9x^3+2x^2-2x-2 [-3, 3, -15, 15]}