15. Mean Value Theorem
( + ) ( )
( 0) = lim ( ) for ( ) = 0 0
provided this limit exists.
Relation to monotonicity: if % (weakly in-
creasing: ( ) ( )) on ( ) and
di erentiable there then 0( ) 0 on ( ).
Proof: As 0 the numerator of ( ) is 0
and continuous, hence ( ) = lim ( ) 0.
(Similarly lim 0 ( ) 0.) ¤
Lab problem: If is di erentiable on ( ) and
attains a maximum (or minimum) at ( )
then ( ) = 0. 82
Mean Value Theorem: If is continuous on [ ]
and di erentiable on ( ) then ( ) with
( ) = ( ) + ( )( )
This is a result of crucial importance in the ap-
proximation of functions.
An interpretation is that di erentiable func-
tions are locally almost linear: If and are
very close, and 0is continuous, we can ap-
proximate 0( ) by 0( ):
( ) ( ) + ( )( );
here the rhs is a straight line (as a function of
), with slope ( ).