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direction parallel to the y -axis) between the graph of the
function f and thechord joining ( a , f ( a )) to ( b , f ( b )). Does D
satisfy the conditions of Rolle's Theorem? What is the result
of applying Rolle's Theorem to D , when expressed in terms of
thefunction f ?
(ii) Let the slope of the chord from ( a , f ( a )) to ( b , f ( b )) be
f ( b ) f ( a )
b a K .
Define the function F :[ a , b ]
R
by
F ( x )
f ( b )
f ( x )
K ( b x ).
Check that F satisfies the conditions of Rolle's Theorem on
[ a , b ].
Apply Rolle's Theorem to the function F , to obtain a value
for K in terms of the function f
.
You should have obtained the same result from qn 13(i) and 13(ii).
This result is called the Mean Value Theorem . Oneapplication is that,
on a continuous and smooth journey, the average speed of the journey
must actually be reached at least once during the journey. While this
result looks obvious enough, it has powerful and significant
applications.
14 Show that the conclusion of the Mean Value Theorem applied to a
function f on an interval [ a , a h ] may be expressed by saying
that, under the appropriate conditions, there is a
, with 0
1,
such that f ( a h ) f ( a ) hf ( a h ).
15 A function f :[ a , b ] R is continuous on the closed interval [ a , b ]
and differentiable on the open interval ( a , b ).
Provethat if f
0 for all values of x in ( a , b ), then the function f
is strictly monotonic increasing.
Show also that if f
( x )
0 for all values of x , then the function f is
strictly monotonic decreasing.
( x )
(16) Functions f
, f
, f
and f
are defined on [0, 2 ]by
x 2 ,
x
cos x
f
( x )
sin x , f
( x )
1
x 6 , f
x 2
x
24
f
( x )
sin x x
( x )
1
cos x .
Prove that each of these four functions is strictly increasing on the
given domain.
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