Let
- Draw the graph of
for
.
- Show that the hypotheses of the mean value theorem are satisfied on
and find the mean values the theorem provides.
- The sketch is as follows:
- Since
and
are continuous on
and
, respectively and are differentiable on
and
, the only point at which this piecewise function might be discontinuous or non differentiable is at
.
At
we have
Thus, the left and right-hand limits are equal, so the limit exists and equals the function value. Therefore,
is continuous at
. Finally, we must check that the derivative exists at
. Since the derivative of
and the derivative of
, both are equal to
at
. Hence, the derivative exists.
Now, we have met the conditions of the mean-value theorem, so we can apply the theorem to conclude,
Further, we know
So,
and