Find two positive numbers such that the sum of the first and twice the second is 56 and whose product is a maximum

Solution:

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

Given, x + 2y = 56

⇒ 2y = 56 - x

⇒ y = 28 - 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 (28 - x/2)

M = 28x - x²/2

On differentiating,

dM/dx = 28 - 2x/2

= 28 - x

For product to be maximum, 28 - x = 0

⇒ x = 28

Substitute x = 28 in x + 2y = 56

⇒ 28 + 2y = 56

⇒ 2y = 28

⇒ y = 14

Therefore, the two positive numbers are 28 and 14.

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Find two positive numbers such that the sum of the first and twice the second is 56 and whose product is a maximum

Summary:

Two positive numbers such the sum of the first and twice the second is 56 and whose product is maximum is x = 28 and y = 14.