What to do if the critical point is not a real number? I have a function and I already differentiate it, but when I put it equals to zero I don't get a real number. What am I doing wrong?
$f(x) = x\sqrt{x^2+1}$
$f'(x) = \sqrt{x^2+1}+\frac{x^2}{\sqrt{x^2+1})}$
$f'(x) = \frac{2x^2+1}{\sqrt{x^2+1}}$
Then I equalizes it to 0
$\frac{2x^2+1}{\sqrt{x^2+1}}=0$
$2x^2+1=0$
$2x^2=-1$
$x^2=-1/2$
So I don't know what to do next. 
When reach the second derivative I get the same problem. What it means?

This function does not have a critical point. Notice that $f'(x)=\frac{2x^{2}+1}{\sqrt{x^{2}+1}}>0$ for all $x\in\mathbb{R}$. So the function $f$ is a  monotonic increasing function. 
If you are doing this for finding maxima of minima, then the answer is maxima or minima do not exist on $\mathbb{R}$. However if you restrict your attention on a compact interval (like $[a,b]$), then the absolute maxima and absolute minima of $f$ on $[a,b]$ are respectively $f(b)$ and $f(a)$. 
