Graph of a function

**vance** · February 22nd 2009, 12:04 AM

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

f(x)=x(2^x)

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!

**running-gag** · February 22nd 2009, 12:57 AM

Hi

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

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

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

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

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

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

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