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(a)

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

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Construct a table of possible values :

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

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

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Observe the table.

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

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(b)

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Consider first number be $\"\"$ .

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Second number be $\"\"$ .

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

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$\"\"$

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

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$\"\"$

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

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$\"\"$

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

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

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$\"\"$

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Equate $\"\"$ to zero:

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$\"\"$

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

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

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$\"\"$

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

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

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