There are always some problems in the text where you are supposed to find the c which ``fits'' the statement of the MVT for a particular function. While that may help you remember the statement, that isn't the point. The point is that the result can be applied theoretically to other facts, as in these corollaries (A ``corollary'' is a theorem that is a simple consequence of an earlier result.):
Corollary 1 If f¢(x) = 0 for all x Î [a,b] , then f is a constant.
Remark 1 Hard to imagine why we need to prove this, isn't it? However, we needed to have essentially this result (the MVT) in order to show this seemingly trivial statement.
Corollary 2 If f¢(x) = g¢(x) for all x Î [a,b] , then, for some constant C , f(x) = g(x)+C .
Remark 2 This statement says that, given a function f(x) , if it has an antiderivative, a function F(x) so that F¢(x) = f(x) , then that antiderivative can be only one function, except possibly for a constant term, because any two antiderivatives of f(x) have f(x) as their derivative, so differ only by a constant. That is, there is only one way to construct an antiderivative, except for a constant being added in, or, looking at it from F(x) 's perspective, F(x) is completely determined by its derivative, except for that constant.
Corollary 3 If f is continuous on [a,b] , and has a positive derivative ( f¢(x) > 0 ), on (a,b) , then f is increasing on the interval (a,b) . If f¢(x) < 0 for all x Î [a,b] , then f is decreasing on (a,b) .
The next result is not really a corollary of the MVT, but is an extension of it. It is used later on in the proof of l'Hôpital's rule.
Theorem 1
[Generalized MVT]. If f(x) and g(x) are continuous
on a closed interval [a,b] , differentiable on the open interval (a,b) ,
and if g¢(x) ¹ 0 for every x Î (a,b) , then there is some point
c Î (a,b) so that
f¢(c)
g¢(c)
=
f(b)-f(a)
g(b)-g(a)
.
Copyright (c) 2000 by David L. Johnson.