Two numbers have a difference of 20. Find the numbers if the sum of their squares is a minimum.

Question

Please help, i cant get the answer! Thank you in advance!

Answer 1

Assuming you are to solve this without using calculus,

x - y = 20 ⇒ y = x - 20

x² + y² is a minimum

x² + (x - 20)² is a minimum.

x² + (x - 20)² = x² + (x² - 40x + 400) = 2x² - 40x + 400

Let f(x) = 2x² - 40x + 400

f(x) is a parabola opening up, so it has a minimum at its vertex, which is at

x = -(-40) / [2(2)] = 10

y = x - 20 = 10 - 20 = -10

Using calculus, you get the same result.