Mean value theorem for vectors I would like some help with the following proof(this is not homework, it is just something that my professor said was true but I would like to see a proof):
If $f:[a,b]\to\mathbb{R}^k$ is continuous and differentiable on $(a,b)$, then there is a $a< d < b$ such that $\|f(b)-f(a)\|\le\|f '(d)\|(b-a)$
Thanks for any help in advance.
EDIT: I forgot to mention that my professor said how it could be proven. He said that one could let u be a unit vector in the direction of f(b)-f(a) and go from there.

Right.  Just define a new function $g(t)=f\bigl( a + t(b-a) \bigr)$ and use the mean value theorem for single-dimensional functions, noting that
$$g'(t) = \frac{\partial f}{\partial x^T}\bigl( a + t(b-a) \bigr) (b-a).$$
