Calculus Optimization problem

Question

Two numbers have a product of 16. Find these numbers if the sum of their squares is to be a minimum.

 

The answers are 4 and 4. But i do not how to get them. Please help

 

 Thankyou

Answer 1

Let x and y be the two numbers.

 

xy = 16

 

Let z = x^2 + y^2

 

We want to minimize z.  Let us make z a function of a single variable, say x, then differentiate z and set the derivative equal to 0, solve for x, and then solve for y:

 

xy = 16 implies y = 16/x

 

z = x^2 + y^2 = x^2 + (16/x)^2 = x^2 + 256 / x^2

 

dz/dx = 2x - 512 / x^3 = 0

 

2x = 512 / x^3

 

2 x^4 = 512

 

x^4 = 512/2 = 256

 

x = 256^(1/4) = 4

 

y = 16/x = 16/4 = 4