Mean Value Theorem is an important generalisation of Rolle's Theorem, which allows us to formally prove our most basic intuition about derivatives, that the sign of the derivative indicates the direction that the function is changing.
By subtracting the secant (f(c)-f(a))x/(c-a) from the function
f, we can get a new function g amenable to Rolle's
Theorem. Finding a turning point b of g, we can infer
that f'(b) will be exactly (f(c)-f(a))/(c-a). This
means that over intervals where f'(b) is always positive,
f(c) will be greater than f(a). Similarly for
negative derivatives, with f(a) < f(c), and zero derivatives, with
f(a) = f(c). This means that if we have two functions
f, and g, with derivatives equal, then
f - g must be a constant, i.e. f' has a unique
antiderivative, up to this constant.
Another intuiton that Mean Value Theorem confirms, is that if we have two
functions f and g, with f(a) <= g(a),
and f'(b) <= g'(b) over the whole interval (a, c),
then f(b) <= g(b) over the whole interval [a, c]. We
shall call this fact the Chasing Theorem, imagining that the function
g is outrunning the function f. We can prove this
easily by noting that g - f will be increasing or constant over
all of [a, c].
Chasing Theorem generalises to higher derivatives, when the values and first
n derivatives of f are initially less than that of
g, and the n+1th derivative of f is less
than that of g over the interval (a, c). The proof of
this is a remarkably simple induction, applying the basic Chasing Theorem to
the nth derivative of f and g, and
repeating, until eventually we get to the base case where n = 1.
This Inductive Chasing Theorem makes for an exceptionally intuitive proof of
Taylor's Theorem.