Science:Math Exam Resources/Courses/MATH 180/December 2017/Question 05 (a)
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Question 05 (a) 

Find the values of the local minima of . 
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 

At local minima, a function has zero derivative. 
Hint 2 

If a function has positive second derivative at a critical point, then this point is a local minimum. 
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. We use the second derivative test, meaning that, according to Hint 1 and Hint 2, we need to find those values where and . Using the power rule, we have and By the factorization of , the candidates for local minima are those satisfying or , namely and . Testing the second derivative at these critical points, we have while Therefore, attains a local minimum at (and a local maximum at ). The only value of the local minima of is . Answer: The only value of the local minima of is . 