Among all pairs of numbers (x,y) such that 8x+2y=34, find the pair for which the sum of squares, x^2+y^2, is minimum.

Question

Among all pairs of numbers (x,y) such that 8x+2y=18, find the pair for which the sum of squares, x^2+y^2, is minimum.

Answer 1

I am going to solve th problem in the large type (not the one in smaller type!).

8x + 2y = 34 => y = 17 - 4x

S = (17 - 4x)² + x²

S = 289 - 136x+ 17x²

dS/dx = 34x - 136 = 0 at the minimum point

x = 4 and y = 1 at the minimum