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Represent any point on the curve by (x, 1-x^2). The distance between (0, 0) and (x, 1-x^2) is
To make this easier, let's minimize the SQUARE of this quantity because when the square root is minimal, its square will be minimal.
So minimize
Find the derivative of L and set it equal to zero.
This gives you
or 
You can use the Second Derivative Test to figure out which value(s) produce the MINIMUM distance.
When x = 0, the second derivative is negative, indicating a relative maximum. When
, the second derivative is positive, indicating a relative MINIMUM.
The two points on the curve closest to the origin are
To make this easier, let's minimize the SQUARE of this quantity because when the square root is minimal, its square will be minimal.
So minimize
Find the derivative of L and set it equal to zero.
This gives you
You can use the Second Derivative Test to figure out which value(s) produce the MINIMUM distance.
When x = 0, the second derivative is negative, indicating a relative maximum. When
The two points on the curve closest to the origin are