Find the numbers whose product is 100 and whose sum is minimum.

Solution:

Given, the product of two numbers = 100

The sum is minimum.

We have to find the numbers.

Let the two numbers be x and y.

xy = 100

So, y = 100/x

Sum = x + y

= x + (100/x)

= f

To find x at which f is maximum

f = x + (100/x)

df/dx = 1 + 100(-1/x²)

= 1 - (100/x²)

d²f/dx² = -100(-2/x³)

= 200/x³

Let df/dx = 0

1 - (100/x²) = 0

1 = 100/x²

x² = 100

Taking square root,

x = ±10

At x = 10,

d²f/dx² = 200/1000 = 1/5

So, x = 10 is a minimum point.

At x = -10,

d²f/dx² = 200/-1000 = -1/5

So, x = -10 is a maximum point.

At x = 10, the sum is minimum.

At x = 10, y = 100/x = 100/10 = 10

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

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Find the numbers whose product is 100 and whose sum is minimum.

Summary:

The numbers whose product is 100 and whose sum is minimum are x = 10 and y = 10.