Question 1209656
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What is the smallest distance between the origin and a point on the graph of y = {{{(1/sqrt(3))*(x^2 - 7 + 2x)}}}?
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<pre>
The square of the distance from the origin to the point (x,y) is  D = x^2 + y^2.


We have 

    D = x^2 + y^2 = x^2 + {{{(1/3)*(x^2+ 2x - 7)^2}}}


I will not simplify this expression.


Instead, I will find the minimum of D over {x} graphically.


I use the plotting tool at website  www.desmos.com/calculator/


It provides the plot (free of charge) and the position and the coordinates of the minimum.


My plot is under this link https://www.desmos.com/calculator/4phmfmwfj7


To see the coordinates of the minimum, click on it.


We have  {{{D[min]}}} = 3.04039  approximately.


Hence, the minimum distance under the problem's question is  {{{sqrt(3.04039)}}} = 1.7437.


<U>ANSWER</U>.  The smallest distance between the origin and a point on the graph 

         of  y = {{{(1/sqrt(3))*(x^2 - 7 + 2x)}}}  is  1.7437, approximately.
</pre>

Solved.