SOLUTION: Among all pairs of numbers (x,y) such that 3x+y=15, find the pair for which the sum of squares, x^2+y^2, is minimum. Write your answers as fractions reduced to lowest terms.
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Question 1150050: Among all pairs of numbers (x,y) such that 3x+y=15, find the pair for which the sum of squares, x^2+y^2, is minimum. Write your answers as fractions reduced to lowest terms. Answer by ikleyn(52776) (Show Source):
y = 15-3x
Hence, x^2 + y^2 = x^2 + (15-3x)^2 = x^2 + 225 - 90x + 9x^2 = 10x^2 - 90x + 225.
A quadratic function ax^2 + bx + c with positive coefficient "a" has the minimum at x = .
In your case, a = 10 and b = -90, hence x = - = .
Then y = = = .
ANSWER. This pair is (x,y) = (,).
