MATHEMATICAL ANALYSIS 09
Consequences of the Mean Value Theorem
The MVT turns information about derivatives into strong conclusions about entire functions.
Zero derivative means constant
If \(f'(x)=0\) throughout an interval, then for any \(a\lt b\), the MVT gives
\[\frac{f(b)-f(a)}{b-a}=f'(c)=0,\]so \(f(a)=f(b)\). Therefore \(f\) is constant.
Positive derivative means increasing
If \(f'(x)\gt0\) throughout an interval, then for \(a\lt b\),
\[f(b)-f(a)=f'(c)(b-a)\gt0.\]Hence \(f(b)\gt f(a)\): the function is strictly increasing.
Equal derivatives
If \(f'=g'\) on an interval, then \((f-g)'=0\). Therefore
\[\boxed{f(x)=g(x)+C}\]
This explains the constant of integration.
Error bounds
If \(|f'(x)|\le M\), then
\[\boxed{|f(b)-f(a)|\le M|b-a|}.\]This controls how rapidly a differentiable function can change.
Why this theorem is central
The MVT provides the logical bridge from local derivative information to global behaviour across an interval.