Minimize cosh(x)+tanh(x)

[Eric78]

Hi, I need to find the Y-coordinate of the extrema of cosh(x)+tanh(x),

Thanks!

[dextercioby]

Hi, I need to find the Y-coordinate of the extrema of cosh(x)+tanh(x),

Thanks!

You should be able to reduce everything to a fourth order algebraic equation with integer coefficents.
The condition for extrema (giving the critical points of the graphic) is
(\cosh x+\tanh x)'=0 which gives an transcedental equation in the hyperbolic functions \cosh x and \sinh x .Substitute the definiton of these functions and then make the substitution \exp x = \lambda .The equation for lambda is as i said and look for its positive roots.

[Eric78]

Yes, I found x^3+x+1=0 where x=sinh(x), but there is no simple root, and I need a quite simple solution...

[Tom Mattson]

I'd hunt the root down in 2 steps:

1. The rational root test (http://www.purplemath.com/modules/rtnlroot.htm)

and if no rational roots then...

2. Newton's Method (http://www.math.sc.edu/cgi-bin/sumcgi/Newton.pl)

[Zurtex]

Here is my attempt at the problem:

y = \cosh x + \tanh x

y \cosh x = \cosh^2 x + \sinh x

y' \cosh x + y \sinh x = 2 \cosh x \sinh x + \cosh \x

y'=0

\cosh^2 x + \sinh x = 2 \cosh x \sinh \x + \cosh \x

\cosh^2 x + (\sinh x - \cosh x) - 2 \cosh x \sinh x = 0

\frac{e^{2x} + 2e^x - 2e^{-2x}}{4} + e^{-x} - \frac{e^{2x} - e^{-2x}}{2} = 0

e^{2x} + 2e^x - e^{-2x} + 4e^{-x} - 2e^{2x} + 2e^{-2x}=0

-e^{2x} + 2e^x + 4e^{-x} + e^{-2x}=0

-e^{4x} + 2e^{3x} + 4e^x + 1=0

Have I made a mistake or something? Looks pretty hard to me (although solvable exactly).

[dextercioby]

Yes, I found x^3+x+1=0 where x=sinh(x), but there is no simple root, and I need a quite simple solution...

You're right.This is the quations u should be getting.There is at least one solution in R(which should be negative and in the interval (-1,0)),but to find all of them u have to apply Cardano's formulas.My guess is the other two are complex.And sinh(x) can take only real values,so the path to finding the extremas is hopefully clear from now.