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I was working through some advanced algebra questions when I stumbled across a question I couldn't quite figure out.

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

I got sqrt2.5 but that was incorrect :(

Jul 11, 2020

#1
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Use https://www.desmos.com/calculator to graph it.

It looks like the least distance from the origin to the curve is where

the parabola crosses the x-axis, at about a shade less than +4.5.

Jul 11, 2020
#2
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No there's something bad wrong.  I must have entered it wrong in desmos.  When x = 4.5 then y does not equal 0.

Guest Jul 11, 2020
#3
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We need to minimize $$\sqrt{x^2 + \frac{1}{\sqrt{2}} (x^2 - 3)}$$

Taking the derivative and setting it to 0, the minimum value is sqrt(3) at x = 1.

Jul 11, 2020