Question 03 (c)
Let on the interval . In this question you will sketch this function.
(c) Determine where ƒ is increasing and where it is decreasing. Does ƒ have any local extrema?
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!
To determine where is increasing and decreasing, we should find the first derivative, its critical points, and do the first derivative test.
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.
Using the quotient rule, we find:
Setting , we can clear the denominator and get:
This equation has infinitely many solutions, but only one which is in the interval [0,π]. The trig ratios and are equal on a 45-45-90 degree triangle, so the angle in the interval [0,π] that solves the above equation is 45 degrees, or in radian measure, .
Next, we check to see if there are any critical points where is undefined. The derivative would be undefined where the denominator is equal to zero, but because is always positive, we don't need to worry about this here.
Thus we have one critical point at . Performing the first derivative test, we get:
Hence f(x) has a maximum at .