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The Problem

Hello! I've scavenged the internet and asked two professors, with no avail. I don't want to ask my professors because he does not want to give us the answer, since he wants to put it on the final.

Anyway, story aside the picture is of problem number 55 from chapter 14.7(Maximum and Minimum values) of the book "Calculus" by Stewart.

In addition to the question given by the textbook, my professor told us to 1) Prove the unique critical points exist and 2)prove there is absolute minimum at this point.

The question was given in class but he said it will be on the final so our class is scrambling to find the solution. Other professors are having trouble with it to (or in hindsight we suspect it might be they just don't want to deal with it since we aren't their students).

We found a website that answers the textbook's question, but not for the questions give by the professor: http://www.slader.com/textbook/9780538497817-stewart-calculus-7th-edition/979/exercises/55/#

Any help would be appreciated. Thank you!

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  • $\begingroup$ Which part exactly are you having trouble with? $\endgroup$
    – WaveX
    Oct 24, 2018 at 21:34
  • $\begingroup$ 1) Prove the unique critical points exist and 2)prove there is absolute minimum at this point. The thing is, I understand the chapter very well, it's this specific problem I'm having trouble understanding in the first place. I appreciate you want me to actually learn it, but I assure you I just need the answer and solution, if I see it done once I'll understand what's going on. $\endgroup$
    – Person
    Oct 25, 2018 at 1:28
  • $\begingroup$ autarkaw.org/2012/09/03/… Perhaps this may be if interest? $\endgroup$
    – WaveX
    Oct 25, 2018 at 2:43
  • $\begingroup$ Yes thank you! How do I mark yours as an answer? $\endgroup$
    – Person
    Oct 25, 2018 at 17:51
  • $\begingroup$ Mine was a comment, so you can't accept it. Plus, it's not my original answer, it's someone else's that made the article, they should have the credit, not me $\endgroup$
    – WaveX
    Oct 25, 2018 at 17:54

1 Answer 1

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Let $f(m, b) = \sum_{i=1}^n (y_i - (m x_i + b))^2$. The goal of the least squares method is to find $m$ and $b$ that minimize this function. (Note that the $y_i$ and $x_i$ are just fixed numbers, not variables.)

You are in a calculus class, so you should know how to minimize a nice function like this. Do you know how to find critical points, and why they are relevant to the above minimization problem? Have you figured out where the two equations (at the end of the question) come from?

More specifically, if you try to compute the critical points of $f$ and find only one point, then you have shown existence and uniqueness.

You should also have learned from your class how to classify critical points (local minima, local maxima, etc.). This will help you show the unique critical point you found is an absolute minimum.

Try to take some time to think about the problem and how it relates to things you've learned in the class, instead of searching for answers.

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  • $\begingroup$ Critical points are where the graph of the function changes from increasing to decreasing or the opposite. That means the exact point where the change occurs is either bottom of the curve, or the top. The first order derivative represence the change in the function, so if the change is zero, that must mean it is a critical point (for a single variable function, otherwise it can be a saddle point too) "Have you figured out where the two equations (at the end of the question) come from?" In short, no. I'm simply looking for the solution to the other two questions I asked. Im out of charactrs $\endgroup$
    – Person
    Oct 25, 2018 at 1:34
  • $\begingroup$ @Person So have you tried to find the critical points? (As you said, this amounts to checking where the first derivatives are zero.) $\endgroup$
    – angryavian
    Oct 25, 2018 at 5:34

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