+1679 What is the smallest distance between the origin and a point on the graph of y = 1/2*(x^2 - 8).
+128578 Let the point we want be (x , (1/2)(x^2 - 8))
Using the square of the distance
D^2 = x^2 + [ (1/2)(x^2 -8) ]^2
Take the derivative and set to 0
2x + (x^2 - 8)(2x) = 0
2x + 2x^3 - 16x = 0
2x^3 - 14x = 0
x^3 - 7x = 0
x( x^2 - 7) = 0
x = 0 (reject)
x^2 - 7 = 0
x = sqrt (7)
The point is ( sqrt 7 , -1/2)
The distance is sqrt [ (sqrt 7)^2 + (-1/2)^2 ] = sqrt [ 7 + 1/4 ] = sqrt (29 ) / 2
