Probably a noobish question, but I've got this problem:
Two graphs: $y=0.5x^2$ and $y=-2+x$. A vertical lines goes through these graphs in points A and B. What's the shortest possible distance between A and B?
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1$\begingroup$ Draw a picture. If the vertical line is $x=t$, then the distance is $(0.5)t^2-(-2+t)$. $\endgroup$– André NicolasMay 3, 2016 at 17:45
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$\begingroup$ Try to find the minimum of $0.5x^2-(-2+x)$ $\endgroup$– drhabMay 3, 2016 at 17:45
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2 Answers
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The distance between two graphs can be denoted by $g(x)=|0.5x^2-(-2+x)|=|0.5x^2-x+2|$.
Now we need to minimize the function $g(x)$.
We can do this by taking the derivative of g(x):
$g'(x)=x-1$
Set this to zero:
$x-1=0 \Rightarrow x=1$
So the shortest distance appear at x=1.
And the shortest distance would be: $0.5*1^2-(-2+1)=1.5$
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You want to minimise $\frac{1}{2}x^2-x+2=\frac{1}{2}(x-1)^2+\frac{3}{2}$. So the answer is $\frac{3}{2}$ achieved when the vertical line is $x=1$.
