If , find the least value of .
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:
Let a and b be two positive quantities, S their sum and P their product; then, from the identity:
, we have
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.
To find the greatest value of a product the sum of whose factors is constant.
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.
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:
Show that where r is any real number.
that is, , which is the desired result.
Find the greatest value of for any real value of x numerically less than a.
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 .
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:
Divide an odd integer into two integral parts whose product is a maximum.
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 .
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 .