Science:Math Exam Resources/Courses/MATH101/April 2014/Question 01 (j)
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Question 01 (j) 

Short Problem. Show your work. No credit will be given for the answer without the correct accompanying work. Let . Find the interval(s) on which ƒ is increasing. 
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 that a function is increasing when its derivative is positive. 
Hint 2 

Consider the Fundamental Theorem of Calculus 
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. Note that is increasing on some interval if and only if on . By the Fundamental theorem of Calculus, we have
Since is always positive, we only need to check when the quadratic polynomial is positive. We can factor this to get This will be positive when both factors are positive or both factors are negative. Both factors are positive when and both factors are negative when . Therefore the function is increasing when or . 