I am trying to write a bifurcation diagram for $f'(x)=r-cosh(x)$ and I am having a little trouble. I also have to show that a saddle-node bifurcation occurs at a critical value of $r$.
So, I know $f(x^*,r_c)=r_c-cosh(x^*)$ and $f_x(x^*,r_c)=sinh(x^*)$. I am having a little trouble finding what $r_c$ is equal to because I have never done these problems with cosine and sine functions involved. Could someone please give me a hint?
Thanks for reading this.