Find two positive numbers whose product is 100 and whose sum is a minimum.

Solution:

Given,

Product = 100

Let the two numbers be x and y.

xy = 100

y = 100/x

f(x, y) = x + y

f(x) = x + 100/x

Minimum of f(x) is obtained at f’(x) = 0

f’(x) = 1 - 100/x² = 0

x = ±10.

x = 10

y = 10

Therefore, the two positive numbers are x = 10 and y = 10.

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Find two positive numbers whose product is 100 and whose sum is a minimum.

Summary:

Two positive numbers whose product is 100 and sum is a minimum are x = 10 and y = 10.