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I have two functions for which I have to prove that they are monotonic for $x\in (-\infty,0]$. The first function is:

$f(x)=\frac{1}{2}\left( 2+x^2-\sqrt{4x^2+x^4} \right)$,

the second function is

$g(x)=x+\sqrt{1+x^2}$ .

EDIT: For now I tried to make a straightforward approach for $x_2<x_1$ assumed that $f(x_2)<f(x_1)$. For $f$ I got

$x_2\left( x_2-\sqrt{4+x_2^2} \right)<x_1\left( x_1-\sqrt{4+x_1^2} \right)$

and very simiarily for $g$:

$x_2+\sqrt{1+x_2^2}<x_1+\sqrt{1+x_1^2}$.

In my opinion for $g$ is is quite obvious that when $x_2<x_1$ the inequality holds (am I right?). For $f$ I do not see that, and I am stuck with it.

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    $\begingroup$ What have you tried this far? You will get significantly better answers on this site if you explain your thought process rather than just post problems. $\endgroup$ – Sean English Aug 5 '15 at 14:27
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    $\begingroup$ Do you know whether the functions are increasing or decreasing on interval $(-\infty,0]$ ? Monotonic could mean either. Do you know a calculus-based test for being monotonic? $\endgroup$ – hardmath Aug 5 '15 at 14:34
  • $\begingroup$ @SE318 I did edit the post. The functions seem to be decreasing towards $-\infty$ $\endgroup$ – Misery Aug 5 '15 at 14:45
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    $\begingroup$ In factoring "x" out from under the radical in rewriting $f(x)$, take care to use $|x|$. This is especially important in that $x \le 0$ on the specified interval. $\endgroup$ – hardmath Aug 5 '15 at 22:16
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For the first one: Write it as $\frac{2}{2 + x^2 + \sqrt{4x^2 + x^4}}$ and notice that $x^2$ is decreasing in interval $(-\infty,0]$. Use facts about compositions and sums of decreasing/increasing functions.

For the second one: Write it as $\frac{1}{\sqrt{1 + x^2} - x}$ and notice that $x^2$ and $-x$ are both decreasing. Proceed as in the first case.

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Hint: You can take the first derivative of these functions, for example: for $x \neq 0$, we have:

$$f'(x) = x + \frac{2(2+x^2)}{\sqrt{4+x^2}} $$

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