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Question
Divide the number 20 into two parts such that their product is maximum
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) ......[From (i)]
= 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 y = 20 – 10 ......[From (i)]
i.e., y = 10
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Let f(x) `=2(x+50/x)`
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∴ by the second derivative test f is minimum at x = `root(5)(2)`
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∴ `x=root(5)(2) "cm" , y = root(5)(2) "cm"`
Hence, rectangle is a square of side `root(5)(2) "cm"`
