Minimize function - nested square roots

Question

I would like to find $z$ which minimizes the below, when $x$ is held at a specific value.

$f(x,z) =\sqrt{\sqrt{x^2 + z^2} - 0.25}$

For example; I would like to find the value of $z$ which minimizes the function when $x = 0.5$

Answer 1

Draw a vertical line on the $xz$ plane, $x=0.5$. On the plane, draw a circle with radius $0.25$.

The minimum distance between the circle and the line, square - rooted, is the minimum value

Answer 2

It looks like the $z$ you are looking for is $0$ if $|x|\geq 0.25$, since the function square root only is increasing and taking the square only gives positive numbers. Similarly, if $|x|<0.25$, we only have to make sure that the pair $(x,z)$ is still in the domain of $f$, hence $z=\sqrt{1/16-x^2}$ is the right $z$.