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# Minimum problem

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Find the minimum value of 2x^2 + 2xy + y^2 - 2x + 2y + 4 - x^2 + 4x over all real numbers $x$ and $y.$

Jun 25, 2022

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Rearrange as: $$(y^2 + 2y) +(x^2 +2x+ 2xy) + 4$$

Consider this as 2 separate equations ($$x^2 +2x+ 2xy$$ and $$y^2 + 2y$$) and a constant. Our goal is to minimize the equations.

Start with $$y^2 + 2y$$. The minimum value of this is -1, and it occurs when $$y = {-b \over 2a} = {-2 \over 2} = -1$$

Now, subbing in $$y = -1$$ into the second equation gives us $$x^2 + 2x - 2x = x^2$$

The minimum value of this, quite obviously, is 0, and it occurs when $$x = 0$$.

This means the minimum value is: $$0 - 1 + 4 = \color{brown}\boxed{3}$$

Jun 26, 2022