He used the following helping lemma 2.5.1
to prove the following mean value theorem 2.5.3
Equivalent to the equation (2.16) is the equation:
$\langle a, f(x)-f(x') - Df(\xi)(x-x') \rangle=0$,
from which we can conclude 2 cases:
- case 1, $f(x)-f(x') - Df(\xi)(x-x') = 0$. It is easy to understand the meaning as in the 1-dimensional situation: we can find a point $\xi$ in between the segment $xx'$ at which the directional derivative in the direction $x-x'$(or, the rate of increase of the function $f$ at the point $\xi$ in this direction) is the same as $f(x)-f(x')$
- case 2, $f(x)-f(x') - Df(\xi)(x-x') \neq 0$ but still orthogonal to $a$.
Questions: Is my interpretation of case 1 good enough? How to interpret case 2? Note that the reason the author used inner product to prove this lemma is to lay the foundation of proving the mean value theorem which uses "norm" in its statement.