be a Function with first derivative 
and
second derivative
.
Then ,
( i )
for all x implies 
is strictly concave upward in (a, b).
( ii )
for
all x implies
is strictly concave downward in (a, b).
The point 
where concave upward changes to concave downward is called inflation point
for
.
The following applet helps you to check the above
properties.
To start the applet select a function /or enter your function and hit New
Function. You can see the graph of
,
&
by checking the appropriate boxes. As you move the slider for x you can see the value of
,
and
.