Find the poin on the graph 0f f(x)=1-x^2 that are closest to0(0,0)

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