What is the smallest distance between the origin and a point on the graph of $y=\dfrac{1}{\sqrt{2}}\left(x^2-3\right)$? 

 Aug 4, 2021

First I extracted easy info by looking at the formula.

It is a concave up parabola.  The axis of symmetry is the y axis (x=0) 

The closest point to the origin would have to be in the part under the x axis.

I decided to use calculus to get the answer.  If you have not done any calculus then there must be a different way to do it.

You will need to tell us if you have not done any calc yet.



I took the 2 points


\((0,0) \qquad and \qquad \left(x,\frac{1}{\sqrt2}(x^2-3)\right)\)


I used the distance formula to get an expression for the distance between them. I called it D

Then I differentiated that with respect to x   to get    dD/dx


I then put that =0 to get the stationary (stat) points.  I knew from the graph that any stat point would give a minumum value.


The stat points I got were x= +/- sqrt2.


Find the y value that goes with that and from that find D.


I drew this to help you see what is happening.  It is interactive, you can move the P point to see that these two answer points give the minimum distance from (0,0)




Any questions?  You are welcome to ask but make you show that you have put some effort in to understand this first.

 Aug 6, 2021

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