how to find maximum value without differentiating let x be positive real number, find max possible value of the expression
$$y = \frac{x^2 + 2 - \sqrt{x^4 + 4}}{x}$$
it can be found by differentiating, but is there no other way of finding it, like using AM $\geq$ GM.
or any other method.
i tried
$$y = x + \frac{2}{x} - \sqrt{x^2 + \frac{4}{x^2}}$$
but it gives nothing

You are on the right track!
To begin with, suppose that $x > 0$. Then we can rearrange the proposed expression as
\begin{align*}
f(x) = x + \frac{2}{x} - \sqrt{x^{2} + \frac{4}{x^{2}}} = x + \frac{2}{x} - \sqrt{\left(x + \frac{2}{x}\right)^{2} - 4}
\end{align*}
If we make the substitution $u = x + \dfrac{2}{x}$, one gets:
\begin{align*}
f(u) = u - \sqrt{u^{2} - 4}
\end{align*}
Now you can apply the derivative method. Analogous approach applies to the case where $x < 0$.
Can you take it from here?
