- 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