The concavity test uses the sign of the second derivative of a twice differentiable function 
on an open
interval 
. The function is convex
(concave upward) on 
iff 
throughout 
. It is concave
(concave downward) on 
iff 
throughout 
(Boyd and Vandenberghe 2004, p. 71).
The strict conditions 
and 
throughout 
imply strict convexity and strict concavity, respectively. An isolated zero of 
is compatible with either strict
condition on the shape of the function. For example,
is strictly convex
on 
, even though its second derivative
is zero at 
.
