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## Main Question or Discussion Point

**[SOLVED] Mathematica does not like hyperbolic functions**

So, consider the equation cosh(x)=n*x

For a given n, the equation has 0, 1, or 2 possible values of x. If n is below the critical value, the equation has no solutions. If n is above the critical value, the equation has two solutions. And if n is exactly the critical value, the equation has one solution. My goal is to use Mathematica to show that the critical value is approximately 1.51.

Theoretically, the line Length[Solve[Cosh[x] == n*x, x]] should give the number of solutions for a given n. Then I can make a table of n's and find the point where the number of solutions goes from 0 to 2.

Unfortunately, I keep getting the error:

Solve::tdep: The equations appear to involve the variables to be solved for in an essentially non-algebraic way.

NSolve has the exact same problem. FindRoot always gives exactly one solution, whether there are zero or two solutions to the equation. Is there a way to make Mathematica more cooperative, or another way to go about this problem? Since the TI-89 can handle this problem (but is too slow to be useful), it seems like Mathematica should be able to as well.

Thanks for your help!