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Thread: Graph of a function

  1. #1
    Junior Member
    Feb 2009

    Post Graph of a function

    List all the asymptotes of the graph of the function and the approximate coordinates of each local extremum.


    I've had a hard time trying to solve this graph , I used my graphing calculator, but it didn't help me.
    Acording to my book, the negative x axis is an asymptote. Can someone explain me how to interpret and graph the function properly? Thanks in advance!
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  2. #2
    MHF Contributor
    Nov 2008

    $\displaystyle f(x) = x\:2^x = x\:e^{2\ln x}$

    $\displaystyle f'(x) = e^{2\ln x} (1 + x \ln 2)$

    Therefore f is decreasing from $\displaystyle -\infty$ to $\displaystyle -\frac{1}{\ln 2}$ and increasing from $\displaystyle -\frac{1}{\ln 2}$ to $\displaystyle +\infty$

    The minimum has coordinates $\displaystyle (-\frac{1}{\ln 2} ; -\frac{1}{e\:\ln 2})$

    The limit in $\displaystyle -\infty$ is 0 which shows that x-axis is an asymptot

    The limit in $\displaystyle +\infty$ is $\displaystyle +\infty$

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