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# Test of Mathematics Solution Subjective 33 - Symmetrical Minima  Test of Mathematics Solution Subjective 33 (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.

## Problem

Let be a fixed odd positive integer. Find the minimum value of , where are non-negative integers and .

## Solution

We have . Therefore we get an equation in where is a constant, precisely .

To minimise, we differentiate w.r.t .

So (for minimum )

That gives us .

But the question tell us that is odd and is an integer. therefore we have to take the closest possible integer value to , which is and .

As already defined, taking to be any one of the above automatically takes the other value.

Therefore the minimum value of is given by  . Test of Mathematics Solution Subjective 33 (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.

## Problem

Let be a fixed odd positive integer. Find the minimum value of , where are non-negative integers and .

## Solution

We have . Therefore we get an equation in where is a constant, precisely .

To minimise, we differentiate w.r.t .

So (for minimum )

That gives us .

But the question tell us that is odd and is an integer. therefore we have to take the closest possible integer value to , which is and .

As already defined, taking to be any one of the above automatically takes the other value.

Therefore the minimum value of is given by  .

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