Graph of a function

• Feb 22nd 2009, 12:04 AM
vance
Graph of a function
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!
• Feb 22nd 2009, 12:57 AM
running-gag
Hi

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

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

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 $\displaystyle +\infty$

http://nsa05.casimages.com/img/2009/...5805782962.jpg