SOLUTION: What is the smallest possible value of the multivariable function f(x,y) = 2x^2 +y^2 -2xy+6x-1

Question 1199429: What is the smallest possible value of the multivariable function
f(x,y) = 2x^2 +y^2 -2xy+6x-1
Answer by ikleyn(52788) (Show Source): You can put this solution on YOUR website!
.
What is the smallest possible value of the multivariable function
f(x,y) = 2x^2 +y^2 -2xy+6x-1
~~~~~~~~~~~~~~~~~

    f(x,y) = 2x^2 + y^2 - 2xy + 6x - 1 = re-group = x^2 + (x^2 -2xy + y^2) + 6x - 1 = 

  = (x^2 + 6x - 1) + (x-y)^2 = (x^2 + 2*3x + 9) - 10 + (x-y)^2 = (x+3)^2 + (x-y)^2 - 10.



From this expression for  f(x,y)  it is seen that the minimum of  f(x,y)  is when

    x = -3, y = -3.



Indeed, then both the quadratic terms  (x+3)^2  and  (x-y)^2  achieve their minimum possible 

values of zero simultaneously, and the minimum value of  f(x,y)  is  -10.    ANSWER

Solved.

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