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.

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## Problem

Suppose is a real valued differentiable function defined on with . Suppose moreover satisfies

Show that for every

## Solution

As the question doesn’t requires 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.