A concave function on an interval 
is a real-valued function 
such that 
is convex on 
(Gradshteyn and Ryzhik 2000). The term concave downward has
the same meaning. The function is strictly concave if 
is strictly convex.
For distinct 
and 
in 
, this means that the function
graph lies strictly above the open line segment
joining the two corresponding points.
For a differentiable function on an open interval 
, concavity is equivalent to the function
graph lying on or below every tangent line. If
is twice differentiable
on 
, the concavity test
is 
for all 
. The strict inequality 
throughout 
is sufficient for strict concavity.
