+0

0
102
2

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.

Aug 18, 2019

#1
+6045
0

$$\text{so we want to minimize \left||x|-(-x^2-3x-2)\right|= \left||x| + x^2+3x+2\right|}$$

$$\text{for x<0}\\ \left||x|+x^2+3x+2\right| = \left|x^2 +2x+2\right|= (x+1)^2+1\\ \text{The minimum of this length occurs at x=-1 and is length 1}\\~\\ \text{for x\geq 0}\\ \left||x|+x^2+3x+2\right| = \left|x^2 +4x+2\right| = \left|(x+2)^2-2\right|\\ \text{Since x>0 this has a minimum at x=0 and is length 2}\\ \text{Thus the smallest possible length is 1 which occurs at x=-1}$$

.
Aug 18, 2019
#2
0

THX SO MUCH!!

Guest Aug 18, 2019