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Since we are given the derivative, it is easy to see that the values of the derivative where it is equal to zero or undefined.
The derivative is equal to zero where and where , so at . The derivative is undefined where so at the points .
Notice that the derivative's denominator is always positive since it is a positive number times a square. So when checking for sign changes, we need only to look at the derivative's numerator, that is, the function (note we could check the denominator as well, but this trick saves a lot of valuable time on an exam!). Let's look at each interval individually.
Case 1:
On this interval, we have that at the point that the numerator is and so the function is increasing on this interval.
Case 2:
On this interval, we have that at the point that the numerator is and so the function is decreasing on this interval.
Case 3:
On this interval, we have that at the point that the numerator is and so the function is decreasing on this interval.
Case 4:
On this interval, we have that at the point that the numerator is and so the function is decreasing on this interval.
Case 5:
On this interval, we have that at the point that the numerator is and so the function is decreasing on this interval.
Case 6:
On this interval, we have that at the point that the numerator is and so the function is increasing on this interval.
This completes the problem.
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