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Calculus 4

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 $d(x)$ .

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