Answer to Question #194331 in Calculus for Moe

Question

Find two positive integers whose product is 100 and whose sum is minimum

Answer 1

Let the two positive numbers be x and y.

So, xy = 100 ...................Equation(1)

and x + y = S (To be minimized)

From Equation (1) ,

y = 100/x ..................Equation(2)

x + 100/x = S

(x² + 100) / x = S

For critical points we let dS/dx = 0

dS/dx = [ 2x² - 1( x² + 100) ] / x²

dS/dx = ( x² - 100 ) / x² .....................Equation(3)

For critical points,

 ( x² - 100 ) / x² = 0

x = 10, -10

Since only positive values of x and y are needed, so we discard the negative value

Hence, x = 10

For the minimum value of sum d²S / dx² > 0 at x = 10.

So differentiating Equation (3) with respect to x, we have

d²S / dx² = [ 2x³ - 2x( x² - 100 ) ] / x⁴

d²S / dx² = 100 / x⁴

At x = 10, d²S / dx² = 0.01 > 0

x= 10

y = 100/10 = 10

Hence, for the minimum value of sum the values of x and y are equal to 10 and 10 respectively.