Numerical method to find $\alpha_0$ where $\alpha x=cosh x$ has exactly one solution.
When doing catenoid problems, I came across these three cases: Different $\alpha$ may lead to 0 or 1, or 2 solutions to $\alpha x=cosh x$, and now I want to find the critical solution where $\alpha_0$ leads to exactly one solution to that function.
I haven't learned numerical analysis and optimization. Could anyone give me some idea of what methods I should use? I'll look it up then. Thanks in advance!
Hint
Probably the simplest is to search for the minimum of function $$f(x)=\frac{\cosh (x)}{x}-\alpha$$ for which $$f'(x)=\frac{x \sinh (x)-\cosh (x)}{x^2}$$ So, we look for the zero of function $$g(x)=x \sinh (x)-\cosh (x)$$ By inspection, the root is between $1$ and $1.5$.
So, start any Newton like method to get it and then deduce $\alpha$.