Consider the general first-order ODE:

x=f(x)

When both f and f are zero at the critical point, the stability is determined by the sign of the first non-vanishing higher derivatives. Apply Taylor series expansions and provide simple functions f(x) to illustrate the following:

(a) If the first non-vanishing higher derivative is even (e.g., f), the point is a saddle point, attracting on one side but repelling on the other.

(b) If that derivative is odd, it follows the same sign rules as f