**Exercise 4.5**

**Question 1**

Let be the required number, then the reciprocal is .

differentiating (1) w.r.t

Set

But only lie in the interval , therefore neglecting .

Now substituting , and in (1)

(a) is as small as possible at x=1

(b) is as large as possible at

**Question 2**

Let and be the two required numbers then

From (1) we have

substituting into (2) to get as a function one singe variable

Differentiating w.r.t.

Now we shall put to find the values of which lies in the interval . Therefore, from (3) we have

Thus lies on the interval

We shall find the values of at the end point of the interval ie and .

a) is maximum at , and at

so

when , then

when , then

Thus we conclude that is as large as possible when one of the number is and other is .

b) is minimum when

So since then .

**Question 3**

Let be the length and breadth of the rectangular feiled then

Let be we area of the rectangular field

By using (1), we have

differentiating w.r.t

substituting , 500 and 1000 into (3)

Thus is maximum when

Substituting x=500 in to (1) we get