Find two numbers whose sum is 24 and whose product is as large as possible.

Solution

Let two numbers be x and 24 - x.
Let P = x(24 - x) = 24x - x^{2 $rightwards double arrow space space space space dP over dx space equals space 24 minus 2 straight x.$
$dP over dx equals space 0 space space space gives space us space space 24 minus 2 straight x space equals space 0 comma space space space space space space rightwards double arrow space space straight x space equals space 12 fraction numerator straight d squared straight P over denominator dx squared end fraction equals negative 2 At space space space straight x space equals space 12 comma space space space space fraction numerator straight d squared straight P over denominator dx squared end fraction space equals negative 2 space less than space 0 therefore space space space straight P space is space maximum space space when space space straight x space equals space 12 space space space space space space space rightwards double arrow space space space numbers space are space 12 comma space 24 minus 12 space straight i. straight e. comma space 12 comma space 12$}

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Application Of Derivatives

Find two numbers whose sum is 24 and whose product is as large as possible.

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