Find two positive numbers such that the sum of the first and twice the second is 100 and their product is as large as possible?

Solution:

Let the first number be x and the second number be y.

Given, x + 2y = 100

⇒ 2y = 100 - x

⇒ y = 50 - x/2

Product, M = xy

Product of two numbers is maximum if their derivative is zero i.e. f(x) = xy = 0.

Put value of y in M = xy

M = x (50 - x/2)

M = 50x - x²/2

On differentiating,

dM/dx = 50 - 2x/2

= 50 - x

For product to be maximum, 50 - x = 0

⇒ x = 50

Substitute x = 50 in x + 2y = 100

⇒ 50 + 2y = 100

⇒ 2y = 50

⇒ y = 25

Therefore, the two positive numbers are 50 and 25.

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Find two positive numbers such that the sum of the first and twice the second is 100 and their product is as large as possible?

Summary:

Two positive numbers such that the sum of the first and twice the second is 100 and their product is as large as possible is x = 50 and y = 25.