Among all the pairs of numbers x y such that 2x y 20 find

Among all the pairs of numbers (x, y) such that 2x + y = 20, find the pair of numbers for which x^2 + y^2 is minimum (as small as possible).

Solution

2x + y = 20.

y = 20 - 2x.

we have the function : x² + y² to be minimum.

x² + y²

x² + (20 - 2x)²

x² + 400 + 4x² - 80x

5x² - 80x + 400

This equation represents parabola.

Minimum value of the parabola is given by: x = -b/2a

  we get x = - (-80)/ 2*5                // \"b\" is the coefficient of \"x\" . \"a\" is the coefficient of x²

x = 80/ 10

x = 8

y = 20 - 2x

y = 20 - 2*8

y = 20 - 16

y = 4

(8, 4) is the pair