Let $E_\mu(x)=\mu e^x$. Show that the family $E_\mu$ undergoes tangential bifurcation at $\mu=1/e$. In particular follow out the following steps:
(a) Plot out the diagonal and the graph of $E_\mu (x)$ for $\mu < 1/e$, $\mu=1/e$, and $\mu>1/e$.
(b) Calculate the partial derivative.
(c) Do the fixed points occur for $\mu<1/e$ or $\mu>1/e$.
As you can see, I was able to execute part (a), but I am entirely confused on how to calculate the partial derivatives, which would lead me to the fixed points in part (c) I believe.
