Find the smallest distance between the origin and a point on the parabola y = x^2 - 2.
+37153 There may be an easier way to do this....but here is one way
Distance between origin (0,0) and (x, x^2 -2) using distance formula
d = sqrt ( x-0)^2 + ( x^2 -2)^2 )
= (x^4 -3x^2 +4)1/2
Now take the derivative to find where the slope of this fxn = 0
1/2 (4x^3 - 6x) / ( sqrt (x^4-3x^2+4) = 0
this = 0 when the numerator = 0 2x^3 - 3x = 0
x (2x^2 -3) = 0 means x = 0 or +- sqrt (3/2)
Plugging these values of x into the original equation x^2 - 2 results in coordinate pairs ( 0, -2) and ( sqrt (3/2) , -0.5) ) (and - sqrt(3/2), - 0.5)
Now, using these pairs and the origin in the distance formula shows the blue pair(s) result in the shortest distance to the origin
sqrt ( sqrt(3/2)2 + (-.5)2 ) = d = sqrt ( 7/4) = sqrt (7 ) /2 = 1.32287 units
