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Math Help - Finding the relative extrema

  1. #1
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    Finding the relative extrema

    y = x^4 - 2x^3 + x + 1

    Now the first thing we have to do is finding the derivative, and that's where I'm having a problem. I can't find the critical number.

    My attempt leads to; 4x^3 - 6x^2 + 1 = 0

    I don't know how to proceed from there to get the critical number.
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  2. #2
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    By the Remainder's Theorem, in which we learned in Algebra, we know that the possible zeroes for a polynoimal is:  \frac { factors \ of \ constant \ term }{ factors \ of \ leading \ coefficient } .

    Therefore, the possible list of zeroes here is:  \{ - \frac {1}{4} , - \frac {1}{2} , - 1 , 1, \frac {1}{2}, \frac {1}{4} \}

    Then just try and find which one would be your zero. I plugged in  \frac {1}{2} and it gave me zero, so that is one of your critical points. Check to see if there are any more critical points.

    By the way, the factorization of f' here is  (x- \frac {1}{2} ) (2)(2x^2-2x-1)=(x- \frac {1}{2} )(4)(x^2-x-1/2)

    That should help.
    Last edited by tttcomrader; November 16th 2009 at 05:32 PM.
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  3. #3
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    Quote Originally Posted by tttcomrader View Post
    By the Remainder's Theorem, in which we learned in Algebra, we know that the possible zeroes for a polynoimal is:  \frac { factors \ of \ constant \ term }{ factors \ of \ leading \ coefficient } .

    Therefore, the possible list of zeroes here is:  \{ - \frac {1}{4} , - \frac {1}{2} , - 1 , 1, \frac {1}{2}, \frac {1}{4} \}

    Then just try and find which one would be your zero. I plugged in  \frac {1}{2} and it gave me zero, so that is one of your critical points.
    I'm sorry, I don't quite understand how you got those numbers. The constant term is the '1' in my derivative, I get that, but where did you get those coefficients? 1, 2, -1, 4, etc.?
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  4. #4
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    Quote Originally Posted by Archduke01 View Post
    y = x^4 - 2x^3 + x + 1

    Now the first thing we have to do is finding the derivative, and that's where I'm having a problem. I can't find the critical number.

    My attempt leads to; 4x^3 - 6x^2 + 1 = 0

    I don't know how to proceed from there to get the critical number.
    See here: http://www.wolframalpha.com/input/?i...5E2+%2B1+%3D+0

    Be sure to click on Show steps.
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