Let $\displaystyle f(x)=6x+\frac{1}{x}$ find:

$\displaystyle f'(x)>0$ for $\displaystyle x \in$ _____?

$\displaystyle f'(x)<0$ for $\displaystyle x \in$ _____?

for the derivative I got $\displaystyle 6-\frac{1}{x^2}$.

I set that to zero to find the critical points, and found $\displaystyle \pm \sqrt{\frac{1}{6}}$

For $\displaystyle f'(x)>0$ I got $\displaystyle (-\infty,-\sqrt{\frac{1}{6}}\ ] \cup \[ \sqrt{\frac{1}{6}},\infty )$

For $\displaystyle f'(x)<0$ I got $\displaystyle [ -\sqrt{\frac{1}{6}},0 ) \cup (0,\sqrt{\frac{1}{6}} ]$

I was told that these are incorrect, and I don't understand why. I even graphed the derivative and this looked to me to be correct. Am I close? Please let me know where I am wrong. I really appreciate it. Thanks.