Science:Math Exam Resources/Courses/MATH101/April 2012/Question 08 (b)
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Question 08 (b) 

Prove that for any positive real number a, we have Hint: First find a simple function ƒ(x) such that for . 
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 hint below. Consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it! 
Hint 

Following up on the hint already given, try a straight line as your simple function ƒ(x). It should be the steepest straight line that fits under between 0 and . 
Checking a solution serves two purposes: helping you if, after having used the hint, 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. Following the hint, we want to find a straight line that is below with minimal error. To do this, we require the straight line to pass through at both and , which are the endpoints of the domain of integration. Since at , we look for a straight line that passes through the origin with formula . Then we put , to find , so . Note that the slope of the line , while the initial slope of is given by , so the linear function is really below in the interval Hence, we can deduce the following inequalities: because and is a strictly increasing function. So by comparison, we obtain because for any . 