Science:Math Exam Resources/Courses/MATH100/December 2010/Question 04 (c)
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Question 04 (c) 

FullSolution Problems. In questions 28, 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. Let NOTE: Another notation for tan^{1}(x) is arctan(x) Determine all critical numbers, open intervals where ƒ is increasing or decreasing, and the xcoordinates of all local maxima or local minima (if any). 
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Hint 

The function has critical points where ƒ'(x) = 0, or where ƒ'(x) does not exist, but ƒ(x) is defined. 
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Solution 

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Please rate my easiness! It's quick and helps everyone guide their studies. The critical points of a function f(x) are all the values where 1. or 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 ). 