Question

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

Short answer

The two numbers are 10 and 10.

Step-by-step solution

Given Data

1)The product of the two numbers is 100.

2)The sum is a minimum.

Determination of the two numbers

It is given that the product of the two numbers is 100.

Let the numbers be x and y. This implies that:

\(\begin{array}{c}xy = 100\\y = \frac{{100}}{x}\end{array}\)

The function which represents the sum of two numbers is:

\(f\left( {x,y} \right) = x + y\)……………(1)

Substitute \(y = \frac{{100}}{x}\)in equation (1)

\(f\left( x \right) = x + \frac{{\left( {100} \right)}}{x}\)

Minimum of \(f\left( x \right)\) is obtained at \(f'\left( x \right) = 0\).

\(f\left( x \right) = x + \frac{{\left( {100} \right)}}{x}\)

\(f'\left( x \right) = 1 - \frac{{\left( {100} \right)}}{{{x^2}}}\)

Substitute \(f'\left( x \right) = 0\) as:

\(\begin{aligned}{c}1 - \frac{{\left( {100} \right)}}{{{x^2}}} &= 0\\\frac{{\left( {100} \right)}}{{{x^2}}} &= 1\\{x^2} &= 100\\x &= \pm 10\end{aligned}\)

Thus, the value of x is 10 because the product is positive.

Therefore,

\(\begin{aligned}{c}y &= \frac{{100}}{x}\\y &= \frac{{100}}{{10}}\\ &= 10\end{aligned}\)

Thus, two positive numbers whose product is 100 and whose sum is a minimum are 10 and 10.