Hello, i_zz_y_ill!
I assume you know to set up Lagrnge multipliers.
Find the local max and minima from origin to curve
Answer are: minimum (0, 0), maximum (3,3)
We want to minimize the distance from point to the origin.
The distance function is: .
For convenience, we can minimize the square of that distance: .
Our function is: .
Set the partial derivatives equal to 0, and solve the system . . .
. .
Equate [4] and [5]: .
Factor: .
Factor: .
From , we have: .
And has no real factors.
Substitute [6] into [3]: .
. . Hence, we have: .
Therefore, the critical points are: .
~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~
I could be wrong, but I think this is the Folium of Descartes.
I has a leafshaped loop in Quadrant 1
. . .
Code:

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