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Thread: how to graph function

  1. #1
    Senior Member euclid2's Avatar
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    how to graph function

    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? how do i graph this? of what degree would this function be? i started by marking of course x=-2,x=6,x=1 on the graph but unsure of what to do now.
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  2. #2
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    I don't think you've got enough information. From what you've been given, your function must be at least a quartic. This means its derivative is at least a cubic and its second derivative is at least a quadratic.

    You are told that there is a local maximum at $\displaystyle x = -2$. So $\displaystyle f'(-2) = 0$ and $\displaystyle f''(-2) < 0$.

    You are told that there is a local maximum at $\displaystyle x = 6$. So $\displaystyle f'(6) = 0$ and $\displaystyle f''(6) < 0$.

    You are told that there is a local minimum at $\displaystyle x = 1$. So $\displaystyle f'(1) = 0$ and $\displaystyle f''(1) > 0$.


    Since $\displaystyle f'(-2) = f'(6) = f'(1) = 0$, that means

    $\displaystyle f'(x) = a(x - 6)(x - 1)(x + 2)$.


    You could expand this, then integrate to find a family of curves for $\displaystyle f(x)$, but unless you are given some points that lie on the curve, you won't be able to evaluate the exact curve.
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  3. #3
    Senior Member euclid2's Avatar
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    so even something like this would work? Imageshack - graphgr.png - Uploaded by nameme

    please excuse the ridiculous graph i'm sure you get the idea
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