Science:Math Exam Resources/Courses/MATH100/December 2018/Question 01 (d)
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Question 01 (d) 

The function has a derivative defined and continuous for all , and critical points only for If we know that and , list all points ( and coordinates) where has a local maximum. 
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 

If a function has a derivative defined and continuous everywhere, then its local maxima and minima must occur at points where the derivative is zero. 
Hint 2 

It might help to sketch a graph of the function, recalling that since the graph has derivative defined and continuous, the derivative can only change sign at the critical points. 
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 

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Please rate my easiness! It's quick and helps everyone guide their studies. Since has a continuous derivative everywhere, the only candidates for the local maxima and minima are its critical points. Note that is larger than and , so by the Mean Value Theorem the function has a local minimum at . This rules out the critical point since by the MVT the function is increasing on the interval . Moreover, since is smaller than and , a similar application of the MVT shows that it is a local minimum. We conclude that the only local maximum is at . 