<s>[INST] Find shortest distance from the parabola $y=x^2-9$ to the origin. 
Find shortest distance from the parabola $y=x^2-9$ to the origin.

First, I find minima of $\sqrt{x^2+(x^2-9)^2}$, so use derivative and ...
Is have an easier way?
 [/INST] Here is an easier way: the shortest distance $r$ is taken at the minimum of 
$r^2 = y^2 + x^2 = y^2 + y + 9$ This gets minimal at $y = -0.5$ (take the derivative or write $r^2 = (y+ 0.5)^2 + 8.75 \ge 8.75 $),  so $r = \sqrt{y^2 + y + 9} \ge  \sqrt{8.75} \simeq 2.9580$.
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