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Question
Divide the number 20 into two parts such that their product is maximum.
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Solution
The given number is 20.
Let x be one part of the number and y be the other part.
∴ x + y = 20
∴ y = (20 - x) ...(i)
The product of two numbers is xy.
∴ f(x) = xy = x(20 - x) = 20x - x2
∴ f'(x) = 20 - 2x and f''(x) = - 2
Consider, f '(x) = 0
∴ 20 - 2x = 0
∴ 20 = 2x
∴ x = 10
For x = 10,
f ''(10) = - 2 < 0
∴ f(x), i.e., product is maximum at x = 10
and 10 + y = 20 ....[from (i)]
i.e., y = 10.
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