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

$$y(x) = x + \frac{2}{x} - \sqrt{x^2 + \frac{4}{x^2}} = x + \frac{2}{x} -\sqrt{(x + \frac{2}{x})^2-4}$$
$$t=x+\frac{2}{x}$$
$$y(t)=t-\sqrt{t^2-4} =\frac{4}{t+\sqrt{t^2-4}}$$
which is clearly monotonicly decreasing.
so we need to find the minimal value of $x+\frac{2}{x}$
by AM-GM:
$$\frac{x+\frac{2}{x}}{2}\geq \sqrt{2}$$
