What is the smallest distance between the origin and a point on the graph of \(y=\dfrac{1}{\sqrt{2}}\left(x^2-3\right)?\)
I got sqrt(5/2), but it was wrong...
+128460 I get the same result .....
Let the point we are looking for be ( x , (x^2 - 3) / sqrt 2 )
We can use the square of the distance formula
D^2 = x^2 + (x^2 - 3)^2 / 2 simplify
D^2 = x^2 + (1/2)(x^4 - 6x^2 + 9)
D^2 = x^2 + (1/2) x^4 - 3x^2 + 9/2
D^2 = (1/2)x^4 - 2x^2 + 9/2 take the derivative of the function and set to 0
2x^3 - 4x = 0
x^3 - 2x = 0 factor
x( x^2 - 2) = 0
Setting the second factor to 0 and solving for x we get x = sqrt 2 or x = -sqrt 2
Either value minimizes the distance....so.....one point is (sqrt 2 , -1/ sqrt 2)
And the distance = sqrt ( 2 + (1/2) = sqrt ( 2.5) = sqrt (5/2) ≈ 1.58
Here's a graph : https://www.desmos.com/calculator/3zo8v8cxyz
