Let us suppose you have a certain function $f(x)$ and you want to find out in which intervals this function is decreasing, constant or increasing. I know you need to follow these steps:
- Find out $f'(x)$.
- Find out the values for which $f'(x)=0$. In other words, we need to find out the zeroes of $f'(x)$. Let's suppose we find out two values which are $x=a$ and $x=b$, and that $a<b$.
- Now we need to choose a random value $r$ from the interval $(-\infty,a]$ or $(-\infty,a)$ (I don't remember exactly) and calculate $f'(x)$ for $x=r$.
- If $f'(x)<0$ for $x=r$, then $f(x)$ is decreasing in the mentioned interval. If $f'(x)=0$ for $x=r$, then $f(x)$ is constant in that interval. If $f'(x)>0$ for $x=r$, then $f(x)$ is increasing in that interval.
- We need to repeat steps 3 and 4 for the other intervals.
Now, what happens if $f'(x)$ doesn't have any zeroes? What should I do?
Example of a (derivative) function that doesn't have any zeroes: $e^x/x$.
Thanks in advance!