11.5 MEAN VALUE THEOREM AND TAYLOR'S FORMULA Using D f as a replacement for f', we guess that the multidimensional analogue of the Mean Value Theorem is f(x) - f

(a) = Df(c)(x-a)

for some c "between" x and

a, i.e., some c E L(x;a), the line segment from a to x.

The following result shows that our guess is wrong for functions f : R n ---t R m when m > l.