The graphs of \(y=|x|\) and \(y=-x^2-3x-2\) are drawn. For every \(x\), a vertical segment connecting these two graphs can be drawn as well. Find the smallest possible length of one of these vertical segments.
+6248 \(\text{so we want to minimize $\left||x|-(-x^2-3x-2)\right|= \left||x| + x^2+3x+2\right|$}\)
\(\text{for $x<0$}\\ \left||x|+x^2+3x+2\right| = \left|x^2 +2x+2\right|= (x+1)^2+1\\ \text{The minimum of this length occurs at $x=-1$ and is length 1}\\~\\ \text{for $x\geq 0$}\\ \left||x|+x^2+3x+2\right| = \left|x^2 +4x+2\right| = \left|(x+2)^2-2\right|\\ \text{Since $x>0$ this has a minimum at $x=0$ and is length 2}\\ \text{Thus the smallest possible length is 1 which occurs at $x=-1$}\)
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