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 – x^{2}

⇒ 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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