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Question 04 (c)
Full-Solution Problems. In questions 2-8, justify your answers and show all your work. If a box is provided, write your final answer there. Simplification of answers is not required unless explicitly requested.
NOTE: Another notation for tan-1(x) is arctan(x)
Determine all critical numbers, open intervals where ƒ is increasing or decreasing, and the x-coordinates of all local maxima or local minima (if any).
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!
The function has critical points where ƒ'(x) = 0, or where ƒ'(x) does not exist, but ƒ(x) is defined.
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.
The critical points of a function f(x) are all the values where
2. does not exist and is defined.
First, we look at the part of the domain of where . In this domain:
We see that there are no values of such that the derivative is 0 or undefined and hence there are no critical points there.
We now move on to the part of the domain where :
Here we see that for . And since is in the part of the domain we are looking at, this is indeed a critical point.
Finally, we need to look at = 1. Remember from part (a) that is defined and even continuous at . To see if the function is differentiable at we compute the limit as approaches 1 from either side. First from the left:
and then from the right:
Since the left and right limits are different we can conclude that the function is not differentiable at , which makes that point a critical point. In conclusion, and are the critical points of the function.
To determine the intervals of increase and decrease we need to take test points of the function on the intervals,
Plugging a value from each interval of into our expressions for the derivative, we get the following
Therefore, is increasing on the intervals and and decreasing on the interval . By the first derivative test, we can see that the function has a local maximum at and a local minimum at .
Therefore the function has a local maximum of 2 (at ) and a local minimum of 1 (at ).