Given $f(x) = x^{3}-3x+5$. How do I find all intervals for which the function is monotonically increasing and decreasing?
I have $f'(x)=3x^{2}-3=0 \Rightarrow x=\pm 1$. And $f''(x)=6x$, so $f''(-1)=-6<0$ and $f''(1)=6>0$, so the function has a minimum and a maximum.
So it is definitely increasing and decreasing on some intervals.
What should I specifically look for here to find the intervals?
There is NO need to calculate $f''$.
$f'(x)=3x^2-3$. When $|x|>1$ then $f'(x)>0$. So $f$ is strictly increasing in $(1,\infty)\cup (-\infty ,-1)$. Again , $f'(x)<0$ for $|x|<1$. That is , $f$ is strictly monotone decreasing in $(-1,1)$.
The function $f(x)$ is increasing whenever the derivative $f'(x)$ is positive, and the function is decreasing whenever the derivative is negative.
You have the correct derivative (though there's a typo - it should be $f'(x)=3x^2-3$, but I assume it's a typo because you have the correct solutions for $f'(x)=0$). Now you just need to figure out where that function is positive and where it is negative.
