Difference between revisions of "Lagrange's Mean Value Theorem"
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Latest revision as of 12:53, 20 February 2024
Lagrange's mean value theorem (often called "the mean value theorem," and abbreviated MVT or LMVT) is considered one of the most important results in real analysis. An elegant proof of the Fundamental Theorem of Calculus can be given using LMVT.
Statement
Let be a continuous function, differentiable on the open interval
. Then there exists some
such that
.
Informally, this says that a differentiable function must at some point grow with instantaneous velocity equal to its average velocity over an interval.
Proof
We reduce the problem to Rolle's theorem by using an auxiliary function.
Consider Note that
By Rolle's theorem, there exists
in
such that
or $
which simplifies to
as desired.
QED