Science:Math Exam Resources/Courses/MATH102/December 2014/Question B 01
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Question B 01 

List the xcoordinates of all the minima, maxima and inflection points of the function 
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 

Use first derivative to identify critical points. 
Hint 2 

Use second derivative test to verify whether the obtained critical points are local maxima or minima and also to find inflection 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. First we calculate by the chain rule. We write the function as where and . The chain rule states that and thus
To determine whether it is a maximum or minimum, we calculate the second derivative using a combination of the product and chain rule. This gives As , we know that is a max. To find the inflection points, we solve . As exponentials are always positive, these occur when or, solving for , when Notice the two points divide the whole domain into three intervals and in each of those intervals, plugging in , and , we have
So change signs on both sides of and . So are inflection points. Answer: The given function one point having its maximum and two inflection points and ; . 