Derivatives and signs
The method of the previous section for deciding whether there is a local maximum or minimum at a critical point by testing ‘‘near-by’’ points is not always convenient. Instead, since we have already had to compute the derivative to find the critical points, we can use information about the derivative to decide. Recall that
- If on an interval, then is increasing on that interval.
- If on an interval, then is decreasing on that interval.
Consider the function Find the intervals on which is increasing and decreasing.
Start by computing Now we need to find when this function is positive
and when it is negative. We will find these values by constructing a sign
table
To do this, solve Factor
Now we can check points between the these points to find when is increasing and decreasing:
From this we can make a sign table for :
Hence is increasing on and and is decreasing on and .