let x be a positive real.Find maximum possible value of y=(x^2+2-(x^4+4)^1/2)/x

Hari Shankar

Since denominator is +ve, we must seek the maximum among x>0

We have $y = \frac{x^2+2 - \sqrt{x^4+4}}{x} = x + \frac{2}{x} - \sqrt{x^2 + \frac{4}{x^2}}$

Let $t = x + \frac{2}{x}$

Then $y = t - \sqrt{t^2-4} = \frac{4}{t+\sqrt{t^2-4}}$

But from AM-GM, $t \ge 2 \sqrt 2$ and hence $t + \sqrt{t^2-4} \ge 2 \left(\sqrt 2 + 1\right)$ and hence $y \le 2 \left(\sqrt 2 -1 \right)$