Finding Relative Extrema of g(x)=x sech(x)

[kari82]

Hello,

can i get help with this?

Find any relative extrema of the function

g(x)=x sech(x)

First i find the derivative
g'(x)=sech(x)(-xtanh(x)+1)

set to 0

sechx=0 can't be solved

-xtanh(x)+1=0
xtanh(x)=1
I know i suppose to solve that to get the critical points and then do the sign test... but i don't know how to solve that equation.


Thanks!

 

[Saladsamurai]

Hi kari82

I don't have much experience with hyperbolic functions, so I don't know how or whether that equation can be solved analytically, but perhaps a graph of the derivative can help provide insight to your question:

http://www.wolframalpha.com/input/?i=d/dx(x*sech(x))

If you scroll down, it even tells you the roots. The fact that they used approximately equal to symbol may imply that there is no analytic solution and only a numerical one. But I really don't know for sure.

 

[kari82]

thanks! I didnt know about that website.. very useful... :-)