Science:Math Exam Resources/Courses/MATH104/December 2011/Question 01 (j)
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Question 01 (j) 

Let and be functions such that and are both continuous and increasing for all real numbers . Which of the following is ALWAYS true?

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 

What do you know about and from the statement of the problem? 
Hint 2 

To test (I), define a function . What is the value of the second derivative of ? You can use a similar method to test II and III. 
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 can consider statements I, II, III individually to which of them are true. Statement I: f + g is Concave Up EverywhereLet Statement I is true if the second derivative of is positive for all real numbers . The second derivative of is However, and must be positive for all real numbers (because and are continuous and increasing for all ), so must be positive for all . Statement I is true. Statement II: f  g is Concave Up EverywhereIf we let then However, could be less than for all real numbers, so may not be positive for all . Statement II is false. Statement III: fg is Concave Up EverywhereIf we let then We know that are both positive for all real numbers, but we don't necessarily have that are necessarily positive. So it is possible that is negative and outweighs the positive term . For example, choose Then is increasing but which is negative for small values of (e.g. when ). Statement III is false. The AnswerThe answer is (A) I only. 