This is a Test of Mathematics Solution Subjective 144 (from ISI Entrance). The book, Test of Mathematics at 10+2 Level is Published by East West Press. This problem book is indispensable for the preparation of I.S.I. B.Stat and B.Math Entrance.
Also visit: I.S.I. & C.M.I. Entrance Course of Cheenta
Suppose is a real valued differentiable function defined on
with
. Suppose moreover
satisfies
Show that for every
As the question doesn't require us to find an exact solution rather just an upper bound, we can easily find it by manipulating the given statement after establishing certain properties of .
We see that for all
which means
is an increasing function.
As the domain is we can say that
for all
.
(as
)
Integrating both sides from 1 to
As we have,
Substituting
Hence Proved.
This is a Test of Mathematics Solution Subjective 144 (from ISI Entrance). The book, Test of Mathematics at 10+2 Level is Published by East West Press. This problem book is indispensable for the preparation of I.S.I. B.Stat and B.Math Entrance.
Also visit: I.S.I. & C.M.I. Entrance Course of Cheenta
Suppose is a real valued differentiable function defined on
with
. Suppose moreover
satisfies
Show that for every
As the question doesn't require us to find an exact solution rather just an upper bound, we can easily find it by manipulating the given statement after establishing certain properties of .
We see that for all
which means
is an increasing function.
As the domain is we can say that
for all
.
(as
)
Integrating both sides from 1 to
As we have,
Substituting
Hence Proved.