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