What is the smallest distance between the origin and a point on the graph of y = x^2 - 8.
+66 I think its when y=0 so \(x^2=8 \\ x=2\sqrt2 \\\),
So the answer is \(\boxed{2\sqrt{2}}\)
+128474 Let ( x, x^2 - 8) be a point on the graph
Using the square of the diistance function we have
d ^2 = (x - 0)^2 + ( x^2 - 8 - 0)^2
d^2 = x^2 + x^4 - 16x^2 + 64
d^2 = x^4 - 15x^2 + 64 take the derivative of this function and set it = 0
4x^3 - 30x = 0
x ( 4x^2 - 30) =0
Solving for the second factor
4x^2 - 30 = 0
x^2 = 30 /4
x = sqrt (30) / 2 and y = (30/4 - 8) = -1/2
So....the minimum distance is sqrt [ 30/4 + 1/4] = sqrt (31) / 2
