be
a continuous function such that f is differentiable on
.
Then there exists a point 
in
such that
The case when
is known
as Rolle's theorem.
In the applet below, you can select a function from the list of functions,
enter your choice of values for 
and
,
and
click any one of the slider buttons. This will show the graph of
,
the secant line (blue) passing through the points
and
, the tangent
line (red) to the graph of
at
a point
, and
the values of
and
.
You are
supposed to move the slider for the point so
that f'(c) becomes equals to
in the interval
, the theorem does not hold.
You are supposed to analyze the reason.