**Question 1:**

If , find the least value of .

**Solution 1:**

Let

will be minimum when will be minimum.

Now, ….call this equation I.

Hence, z will be maximum when is maximum but is the product of two factors whose sum is .

Hence, will be maximum when both these factors are equal, that is, when

. From equation I, maximum value of . Hence, the least value of .

**Some basics related to maximum and minimum:**

*Basic 1:*

Let a and b be two positive quantities, S their sum and P their product; then, from the identity:

, we have

and .

Hence, if S is given, P is greatest when ; and if P is given, S is least when . That is, *if the sum of two positive quantities is given, their product is greatest when they are equal; and, if the product of two positive quantities is given, their sum is least when they are equal.*

*Basic 2:*

*To find the greatest value of a product the sum of whose factors is constant.*

*Solution 2:*

Let there be n factors , and suppose that their sum is constant and equal to s.

Consider the product , and suppose that a and b are any two unequal factors. If we replace the two unequal factors a and b by the two equal factors , the product is increased, while the sum remains unaltered; hence, *so long as the product contains two unequal factors it can be increased without altering the sum of the factors; *therefore, the product is greatest when all the factors are equal. In this case, the value of each of the n factors is , and the greatest value of the product is , or .

*Corollary to Basic 2:*

If are unequal, ;

that is, .

By an extension of the meaning of the terms *arithmetic mean *and *geometric mean*, this result is usually stated as follows: *the arithmetic mean of any number of positive quantities is greater than the geometric mean.*

*Basic 3:*

*To find the greatest value of * *when * *is constant; **m,n, p, ….being positive integers.*

*Solution to Basic 3:*

Since m,n,p, …are constants, the expression will be greatest when is greatest. But, this last expression is the product of factors whose sum is , or , and therefor constant. Hence, will be greatest when the factors are all equal, that is, when

Thus, the greatest value is .

**Some examples using the above techniques:**

**Example 1:**

Show that where r is any real number.

**Solution 1:**

Since

Hence,

that is, , which is the desired result.

**Example 2:**

Find the greatest value of for any real value of x numerically less than a.

**Solution 2:**

The given expression is greatest when is greatest; but, the sum of the factors of this expression is , that is, ; hence, is greatest when , that is, . Thus, the greatest value is .

*Some remarks/observations:*

The determination of **maximum **and **minimum** values may often be more simply effected by the solution of a quadratic equation than by the foregoing methods. For example:

*Question:*

Divide an odd integer into two integral parts whose product is a maximum.

*Answer:*

Let an odd integer be represented as ; the two parts by x and ; and the product by y; then ; hence,

but the quantity under the radical sign must be positive, and therefore y cannot be greater than , or, ; and since y is integral its greatest value must be ; in which case , or n; thus, the two parts are n and .

**Sometimes we may use the following method:**

Find the minimum value of .

**Solution:**

Put ; then the expression

which in turn equals

.

Hence, the expression is a a minimum when the square term is zero; that is when .

Thus, the minimum value is , and the corresponding value of x is .

**Problems for Practice:**

- Find the greatest value of x in order that may be greater than .
- Find the minimum value of , and the maximum value of .
- Show that and .
- Find the maximum value of when x lies between 7 and -2.
- Find the minimum value of .

More later,

Nalin Pithwa.