Real Analysis Extrema problem

allison713

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Real Analysis Extrema problem (solved)

The problem as it is listed in the book:
problem 4.JPG
And this is my attempt at working the problem. My idea was to find the derivative of the sum and set that equal to zero to identify an extrema and then test if it is the minimum. However, I end up with a value for x that depends on n. Also the problem doesn't define a except that it is made up of real numbers, which could be any real numbers.
Problem4.jpg
 
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Yes, obviously the extreme value of a function will depend on the function! In particular, for this function it will depend upon the values of all the "\(\displaystyle a_n\)"s as well as n itself.
 
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The problem as it is listed in the book:
View attachment 4626
And this is my attempt at working the problem. My idea was to find the derivative of the sum and set that equal to zero to identify an extrema and then test if it is the minimum. However, I end up with a value for x that depends on n. Also the problem doesn't define a except that it is made up of real numbers, which could be any real numbers.
Question: did you establish that critical point is actually a relative minimum?

What are you to say about uniqueness?
 
Okay so the idea is that I just need to find the critical point relative to the function. So it is going to be a function of n, not an actual number. I for some reason kept thinking it needed to be an actual point. That was dumb.
Untitled-1.jpg

Thank you both for your help.
 
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