Science:Math Exam Resources/Courses/MATH110/April 2016/Question 05 (f)
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Question 05 (f) 

Let (f) Find all local maximum and minimum values of , if they exist. 
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 hints below. Read the first one and consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it! If after a while you are still stuck, go for the next hint. 
Hint 1 

Recall the first derivative test for local extremes; Let be a continuous function with or is undefined. (i) If (left of ) and (right of ) , then is a local maximum (i) If (left of ) and (right of ) , then is a local minimum. 
Hint 2 

(For the alternative solution) Recall the second derivative test for extremes: (1) If and , then has a local maximum at ; (2) If and , then has a local minimum at ; 
Checking a solution serves two purposes: helping you if, after having used all the hints, you still are stuck on the problem; or if you have solved the problem and would like to check your work.

Solution 1 

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Please rate my easiness! It's quick and helps everyone guide their studies. From the part (a) and (d) of this question, we know that the domain of is and the derivative of is Then, we can easily see that the only critical point of is (because . Note that is excluded because it is not in the domain.) Since the local extreme(s) only occur at the critical point(s), is the only candidate for the local extreme. In the part (e), we observe that is increasing on , while it is decreasing on , by the first derivative test in the Hint, the point is the local maximum. 
Solution 2 

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Please rate my easiness! It's quick and helps everyone guide their studies. (Alternative solution) From the solution 1, we know that the only critical point is . Now let’s calculate the second derivative At , , so by the second derivative test in the Hint 2, it attains local maximum at this point. In all, it has no local minimum, but has a local maximum at and . 