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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

Artur Wiadrowski
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hemant kumar
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  • In what you tried you assumed $;x>0;$ , as you put the $;x;$ in the denominator inside the square root. Is this so? – DonAntonio Aug 23 '22 at 16:45
  • yes,i did that, but doesn't seem to work – hemant kumar Aug 23 '22 at 16:47
  • The function has no maximal value. This follows from the fact that it behaves "almost" as a straight line with slope 1 . Yet if you assume $;x>0;$ , as apparently you did, it does have a maximal value – DonAntonio Aug 23 '22 at 17:13
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    @DonAntonio x is a positive number as mentioned i the question. Btw how did you assumed the function to be almost straight line. – hemant kumar Aug 23 '22 at 17:16
  • You can graph the function, or else you can that its limit wne $;x\to0;$ is zero, and the limit of $;f(x)-x;$, with $;f;$ your function, is a constant number (in fact, $;-1/2;$ , I believe...) – DonAntonio Aug 23 '22 at 17:30

2 Answers2

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

razivo
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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?

Átila Correia
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