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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
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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)

 

https://www.geogebra.org/classic/h3qg8gay

 

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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