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Question
Find two numbers whose sum is 24 and whose product is as large as possible.
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Solution
Let first number = x then second number = 24 - x.
According to the question, their product p = x(24 - x) = 24x - x2 ..…(1)
For highest and lowest value, `(dp)/dx = 0`
On differentiating equation (1) with respect to x,
`(dp)/dx = 24 - 2x`
`=> 0 = 24 - 2x`
`=> 2x = 24`
` therefore x = 12`
Again differentiating equation (1) with respect to x,
`(d^2p)/(dx^2) = - 2` (negative value)
`((d^2p)/dx^2)_(x = 12) = -2 < 0`
p has a masimum value at x = 12
So, the requied number are 12 and 24 - 12 = 12
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