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Math Help - Quick Polynomial Function Question

  1. #1
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    Quick Polynomial Function Question

    For the function
    f(x) = x^{3} - x^{2}=4x-3

    I find the derivative to be:
    f'(x) = 3x^{2}-2x+4

    and I need to set the derivative to equal 0 for this question I'm doing, I would guess I need to use the quadratic formula to do this (if this is wrong please let me know)

    But when I use the quadratic formula I get this far:
     <br />
x = \frac{2 \pm \sqrt{4-48}}{6}<br />

    which gives me:
     <br />
x = \frac{2 \pm \sqrt{-44}}{6}<br />

    Which can't be right since you can't take the sqrt of a negative number....please help!
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  2. #2
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    Quote Originally Posted by bearhug View Post
    For the function
    f(x) = x^{3} - x^{2}=4x-3

    I find the derivative to be:
    f'(x) = 3x^{2}-2x+4

    and I need to set the derivative to equal 0 for this question I'm doing, I would guess I need to use the quadratic formula to do this (if this is wrong please let me know)

    But when I use the quadratic formula I get this far:
     <br />
x = \frac{2 \pm \sqrt{4-48}}{6}<br />

    which gives me:
     <br />
x = \frac{2 \pm \sqrt{-44}}{6}<br />

    Which can't be right since you can't take the sqrt of a negative number....please help!
    If you mean:

    f(x) = x^{3} - x^{2}+4x-3

    then all your considerations and calculations are OK.

    This function doen't have any turning points, only one point of inflection.
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  3. #3
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    Quote Originally Posted by bearhug View Post
    Which can't be right...
    It is right.

    ...since you can't take the sqrt of a negative number
    This is correct when dealing with real numbers. Since your radicand is negative, f'(x)=0 has no real roots.

    It might help if you post the actual question you are dealing with.
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  4. #4
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    the actual question is:

    for the function f(x) = x^{3} - x^{2}+4x-3, determine:
    a) the intervals of increase or decrease
    b) the location of any max or min points
    c) the intervals of concavity up or down
    d) the location of any points of inflection

    I've done other questions like this but I've always gotten the derivative to equal 0, so I never had any problems.

    for (a) I need set the derivative to equal 0 so I can figure out the x's and know which intervals I need to look at to see where the function is increasing and decreasing.
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  5. #5
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    Quote Originally Posted by bearhug View Post
    the actual question is:

    for the function f(x) = x^{3} - x^{2}+4x-3, determine:
    a) the intervals of increase or decrease
    b) the location of any max or min points
    c) the intervals of concavity up or down
    d) the location of any points of inflection

    I've done other questions like this but I've always gotten the derivative to equal 0, so I never had any problems.

    for (a) I need set the derivative to equal 0 so I can figure out the x's and know which intervals I need to look at to see where the function is increasing and decreasing.
    The function is increasing over (-oo, +oo), as your previous calculations suggest. This also answers (b).

    To answer (c) and (d) you first need to solve f''(x) = 0.
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  6. #6
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    Quote Originally Posted by bearhug View Post
    for (a) I need set the derivative to equal 0 so I can figure out the x's and know which intervals I need to look at to see where the function is increasing and decreasing.
    We have told you: the derivative is never zero, because the equation f'(x)=0 has no real solutions. So your interval is the entire real line, (-\infty,\,\infty).
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  7. #7
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    Ah I get it!
    Thank you all so much.
    I can do (c) and (d) easy, (a) and (b) were just a bit confusing to me.
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