Science:Math Exam Resources/Courses/MATH101/April 2010/Question 09
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For any real number , define . Find the minimum value of .
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We can consider as a constant inside the integral
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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:
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