Exercise 13.  Consider the differentiable function    and the two points  .  

Show that there does not exist a point  c  on the line    between    such that  .  

This result shows that the mean value theorem for derivatives does not extend to complex functions.  

Solution 13.

See text and/or instructor's solution manual.

Solution.  Computing the difference quotient we obtain

        .

The minimum modulus of points    on the line   occurs at the point      and   .

But  ,  and the solutions to    are    and they have moduli  ,  for  .  

And we observe that  

           

Thus    for points     on the line joining   .

Therefore, we conclude that there cannot exist  point  c  on the line    between    such that  .  

This solution is complements of the authors.

(c) 2008 John H. Mathews, Russell W. Howell