If $$x$$ and $$y$$ are real numbers, the least possible value of the expression $$4(x - 2)^{2} + 4(y - 3)^{2} - 2(x

1. If $$x$$ and $$y$$ are real numbers, the least possible value of the expression $$4(x - 2)^{2} + 4(y - 3)^{2} - 2(x - 3)^{2}$$ is :

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By: anil on 05 May 2019 02.35 am

$$4(x - 2)^{2} + 4(y - 3)^{2} - 2(x - 3)^{2}$$
$$y$$ is an independent variable. The value of $$y$$ is unaffected by the value of $$x$$. Therefore, the least value that the expression $$4(y-3)^2$$ can take is $$0$$ (at $$y=3$$).

Let us expand the remaining terms.
$$4(x-2)^2-2(x-3)^2 = 4x^2+16-8x-2x^2-18+12x$$
$$=2x^2+4x-2$$
=$$2(x^2+2x-1)$$
=$$2(x^2+2x+1-2)$$
=$$2((x+1)^2-2$$
The least value that the expression $$(x+1)^2$$ can take is $$0$$ (at $$x$$ = $$-1$$)
Therefore, the least value that the expression $$2((x+1)^2-2$$ can take is $$2*(0-2)=2*(-2) = -4$$
Therefore, option B is the right answer.

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