Q. 134.4( 42 Votes )

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

Answer :

Let one number be x. Then, the other number is (24 –x).

Let P(x) denote the product of the two numbers.


Then, we get,


P(x) = x(24 –x) = 24x – x2


P’(x) = 24 – 2x


P’’(x) = -2


Now, P’(x) = 0


x = 12


And


P’’(12) = -2 <0


Then, by second derivative test,


x = 12 is the point of local maxima of P.


Therefore, the product of the numbers is the maximum when the numbers are 12 and 24 – 12 = 12.


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