Science:Math Exam Resources/Courses/MATH101/April 2010/Question 09
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Question 09 

For any real number , define . Find the minimum value of . 
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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 

We can consider as a constant inside the integral 
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Solution 

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Please rate my easiness! It's fun and helps everyone guide their studies. First we solve the integral considering as a constant:
The first integral in the sum is
Now we solve by parts letting and so that and . Hence
The last integral in the sum is:
So
The derivative is directly computed to be:
Which is zero at . To make sure it is a minimum we use the second derivative test. The second derivative is:
Which is always positive (Remember that ). This means is the minimum for . Plugging in this value gives

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