In my textbook, there is an alternative perspective on the mean value theorem that I don't understand. When we introduced the mean value theorem the first time, the statement looked like this:
Now, the author wants to introduce the mean value theorem in the n-dimensional space, and he begins like this:
The mean value theorem of a function of one variable can be written like this: Assume $f: I \rightarrow \Bbb R$ is differentiable on I $\subset \Bbb R$ and $x, x + \xi \in I$, then there is a $\lambda \in [0, 1]$ such that
$$f(x + \xi) - f(x) = f'(x + \lambda \xi) * \xi$$
He doesn't give a proof on this, and there are a few things that I don't comprehend.
Why is there a $\xi$ on the right side of the equation? (outside of the parenthesis) What does this give me?
What does $f(x + \xi) - f(x)$ tell me? Shall I comprehend this as the slope of the secant that connects $f(x + \xi)$ and $f(x)$?