Is #f(x)=(x-1/x)# concave or convex at #x=-1#?

1 Answer
Feb 10, 2017

Answer:

Since #f''(-1)=2>0#, the function is convex (commonly called concave up) at #x=-1#.

Explanation:

The convexity and concavity of a function can be determined through its second derivative. At #x=a#, a function #f# is:

  • convex (commonly known as concave up) if #f''(a)>0#
  • concave (commonly known as concave down) if #f''(a)<0#

Find the function's second derivative by rewriting with a negative power then using the power rule:

#f(x)=x-1/x#

#f(x)=x-x^-1#

#f'(x)=1-(-1x^-2)#

#f'(x)=1+x^-2#

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

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

The value of the second derivative at #x=-1# is:

#f''(-1)=-2/(-1)^3=-2/(-1)=2#

Since #f''(-1)=2>0#, the function is convex (commonly called concave up) at #x=-1#.