Conjugates of norms

How would one find the conjugate of the following : $$f(x) = \|x\|^2 /2$$ The conjugate function is defined as $ f^*(y) = \max_x y^Tx - f(x)$

I am stuck at how I can derive the explicit form for $x$.

So far, here are my steps:

To maximize I take the derivative and set to $0$.

$$f'(x) = y - \partial\|x\| \cdot \|x\| = 0$$

$$\partial\|x\| = y/\|x\| $$

Edit : $\|x\|$ is any norm here. Not just the 2-norm.

Where do I go from here?

For norm, conjugate is indicator of unit ball for dual norm.

See Conjugate Functions at page 7-9:

By Lieven Vandenberghe.

Edit: I transcribed a proof from Example 3.27 (pp. 93-94) of Boyd and Vandenberghe here.

Here is a proof in the special case that $\| \cdot \|$ is the $\ell_2$-norm.

Note that $\nabla f(x) = x$. When you set the gradient equal to $0$, you get $y - x = 0$, or $x = y$. Thus $f^*(y) = y^T y - \|y\|^2/2 = \|y\|^2/2$.