What is the shortest distance from the point $(0, 2)$ to the parabola $y = x^{2}$?

Explanation

1. 01

   A point on the parabola is $(x, x^{2})$; the square of its distance from $(0, 2)$ is $D = x^{2} + (x^{2} - 2)^{2} = x^{4} - 3x^{2} + 4$.

2. 02

   $D' = 4x^{3} - 6x = 0$ gives $x^{2} = \frac{3}{2}$, where $D = \frac{9}{4} - \frac{9}{2} + 4 = \frac{7}{4}$; the distance is $\frac{\sqrt{7}}{2}$.