Question 05 (e)
(e) Determine where ƒ is increasing and where it is decreasing.
Make sure you understand the problem fully: What is the question asking you to do? Are there specific conditions or constraints that you should take note of? How will you know if your answer is correct from your work only? Can you rephrase the question in your own words in a way that makes sense to you?
If you are stuck, check the hint below. Consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it!
Whether f is increasing or decreasing is related to the sign of its first derivative, f' .
Checking a solution serves two purposes: helping you if, after having used the hint, you still are stuck on the problem; or if you have solved the problem and would like to check your work.
We first find the derivative of f(x). Using the quotient rule,
Since the denominator is always positive on the domain (squaring always yields a positive number), we only need check whether the numerator is positive or negative. When , is positive and when , is negative. Thus f is increasing on the interval and decreasing on the interval .