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

1 Answer
Nov 29, 2016

Since #f''(-1)>0#, we see that #sinx# is convex ("concave up") at #x=-1#.

Explanation:

We have to know that #d/dxsinx=cosx# and #d/dxcosx=-sinx#.

Also recall that concavity and convexity are determined through the sign of the second derivative of a function.

First finding the second derivative:

#f(x)=sinx#

#f'(x)=cosx#

#f''(x)=-sinx#

If #f''(-1)>0#, then #f# is convex (commonly called "concave up") at #x=-1#.

If #f''(-1)<0#, then #f# is concave (commonly called "concave down") at #x=-1#.

We see that

#f''(-1)=-sin(-1)approx08415#

Since #f''(-1)>0#, we see that #sinx# is convex ("concave up") at #x=-1#.