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Thread: graph defined?

  1. #1
    Junior Member
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    graph defined?

    Hi,

    I am attempting to graph $\displaystyle (x^2-3)^\frac{2}{3}$.

    My textbook shows the graph as being defined between $\displaystyle \sqrt{3}$ and $\displaystyle -\sqrt{3}$ in fact the derivative indicates that the graph has a local maximum at $\displaystyle x=0$. In my textbook the graph is convex and positive between $\displaystyle \pm\sqrt{3}$

    The problem is that $\displaystyle f(0)=-3^\frac{2}{3}$ is approximately $\displaystyle -2.08$. Which means it's negative, unless I am calculating $\displaystyle f(0)$ incorrectly? What has me really confused is that MuPAD does not show the graph as defined between $\displaystyle \sqrt{3}$ and $\displaystyle -\sqrt{3}$ which has got me thinking the textbook might be wrong. When calculating $\displaystyle -3^\frac{2}{3}$ do you square the minus three and then take a cube root? Maybe that's what's wrong. Maybe f(0) is actually positive?

    Regards
    Craig.
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  2. #2
    Member
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    Re-writing f as

    $\displaystyle (\sqrt[3]{x^2-3})^2$ should hopefully convince you that for real x this is always positive.
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  3. #3
    Junior Member
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    Calculators don't work...

    Hi,

    Thanks, looking at my notes that what I originally thought.
    Does that mean that $\displaystyle f(0)=(-3)^\frac{2}{3}$ is approximately positive 2.08? My calculator gives $\displaystyle (\sqrt[3]{3}[\frac{1+i\sqrt{3}}{2}])^2$ which blows my pip. Why all the complex numbers if it's 2.08?
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  4. #4
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    Cube rooting is defined all along the real line; no complex numbers should be involved at all. Yes f(0) is approximately +2.08.
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