A point P(x,x^2) lies on the curve y=x^2. Calculate the minimum distance from the point A(2,-1/2) to the point P.

The distance between points (x,x^2) and (2,-1/2)

is d(x):=\sqrt{(x-2)^2+(x^2+1/2)^2}

Recalling that t\mapsto \sqrt{t} is strictly increasing, minimize
e(x):= (x-2)^2+(x^2+1/2)^2 .

e(x)=x^2-4x+4+x^4+x^2+1/4 = x^4+ 2x^2 -4x +17/4
e'(x)=4x^3+4x-4=4(x^3+x-1)

Imposing e'(x)= 4(x^3+x-1) =0 you get values for the optimal x
then just substituting this value in the expression of d(x).