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Math Help - A function has a local maximum at x=-2, 6, and a local minimum at x=1.

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    A function has a local maximum at x=-2, 6, and a local minimum at x=1.

    I can't solve this question, I need some help! Please sketch the graph if you can!

    A function has a local maximum at x=-2, a local maximum at x=6, and a local minimum at x=1. What does this information tell you about the function? Sketch a possible function that has these characteristics.
    Last edited by Chris L T521; January 5th 2011 at 07:18 AM. Reason: Restored Original Post.
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    What are the derivatives at those locations?
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    Quote Originally Posted by dwsmith View Post
    What are the derivatives at those locations?
    Sorry, I updated my post
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    Quote Originally Posted by razr View Post
    Sorry, I updated my post
    The question still stands.
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    f'(-2)=f'(6)=f'(1)=0

    f'(x)=a(x+2)b(x-6)c(x-1)=abcx^3-5abcx^2-8abcx+12abc \ \ a,b,c\in\mathbb{R}

    \displaystyle f(x)=\int f'(x)dx
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    Notice that the problem says "sketch a possible function". There are an infinite number of such functions, you should try graphing the simplest one. What does a graph look around a minimum or maximum? Notice that you are not given the value of the function at x= 1, -2, or 6 so you can take them to be whatever you want. (The "local minimum" can be higher than the "local maxima"!)
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    Note that the derivatives at these points can also be infinite of undefined. If the derivative is infinite there is a cusp. If the derivative is undefined there is a sharp edge. (There are other possibilities for undefined derivatives, but in the other cases there would be no min or max.)
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