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