\

(a)

\

Find two numbers whose sum is $\"\"$ and whose product is a maximum

\

Construct a table of possible values :

\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \

First Number ( $\"\"$ )	Second Number ( $\"\"$ )	Product ( $\"\"$ )
$\"\"$	$\"\"$	$\"\"$
$\"\"$	$\"\"$	$\"\"$
$\"\"$	$\"\"$	$\"\"$
$\"\"$	$\"\"$	$\"\"$
$\"\"$	$\"\"$	$\"\"$
$\"\"$	$\"\"$	$\"\"$
$\"\"$	$\"\"$	$\"\"$
$\"\"$	$\"\"$	$\"\"$
$\"\"$	$\"\"$	$\"\"$
$\"\"$	$\"\"$	$\"\"$
$\"\"$	$\"\"$	$\"\"$

\

Observe the table.

\

The numbers are $\"\"$ , $\"\"$ then the product is maximum.

\

\

(b)

\

Consider first number be $\"\"$ .

\

Second number be $\"\"$ .

\

The sum of the two number is $\"\"$ :

\

$\"\"$

\

Let $\"\"$ be the product of two numbers then $\"\"$ .

\

$\"\"$

\

Substitute value of $\"\"$ in $\"\"$ .

\

$\"\"$

\

This product has the maximum value at a point where $\"\"$ .

\

Differentiate $\"\"$ with respect to $\"\"$ :

\

$\"\"$

\

Equate $\"\"$ to zero:

\

$\"\"$

\

This is a maximum value, since $\"\"$ and $\"\"$

\

Substitute the $\"\"$ value in $\"\"$ :

\

$\"\"$

\

The product value is maximum when the two numbers are $\"\"$ , $\"\"$ .

\

\

(a) $\"\"$ , $\"\"$ .

\

(b) $\"\"$ , $\"\"$ .