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Math Help - Find the minimum value of..

  1. #1
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    Wink Find the minimum value of..

    Please help me with this question...

    Find min value of  f(x)=(x-2006)(x-2007)(x-2008) (x-2009)

    Thanks..
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  2. #2
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    Re: Find the minimum value of..

     f(x)=(x-2006)(x-2007)(x-2008) (x-2009)
    Multiply it all out, take the derivative, set it equal to 0 and solve for x.

    Take the 2nd derivative and evaluate it at each of these points.

    Doing this you find 3 stationary points. 2 are minima, 1 is a local maxima.

    You can work out the details.
    Last edited by romsek; February 4th 2014 at 07:55 AM.
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  3. #3
    MHF Contributor ebaines's Avatar
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    Re: Find the minimum value of..

    Since this equation is a 4th order polynomial, the derivative is a cubic which will be difficult to factor. One way to make this easier is to shift the values to a range that is easier to work. For example if you let w = x-2007.5, then finding the min of (w-3/2)(w-1/2)(w+1/2)(w+3/2) = w^4 -5/2w^2+9/16 is much easier. Solve that, then add 2007.5 to get the answer to this problem.
    Last edited by ebaines; February 4th 2014 at 12:48 PM.
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  4. #4
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    Re: Find the minimum value of..

    "This problem" asks for the maximum value of f. It is not necessary to "add 2007.5". Evaluating w^4- (5/2)w^2+ 9/16 at the value of w that gives a maximum will give the maximum value of f. Adding 2007.5 to that value of w will give the corresponding value of x.
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  5. #5
    MHF Contributor ebaines's Avatar
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    Re: Find the minimum value of..

    You're right - I was thinking that the value of x was needed, but it's not. Thanks.
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  6. #6
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    Re: Find the minimum value of..

    Quote Originally Posted by Deci View Post
    Please help me with this question...
    Find min value of  f(x)=(x-2006)(x-2007)(x-2008) (x-2009)
    Look at this plot,
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