Exercise 104

Derive an analogous classification of critical points for equations in one dimension, such as $x'= f(x)$ based on the derivative. A point $x_0$ is critical when $f(x_0) = 0$ and almost linear if in addition $f'(x_0) \neq 0$. Figure out if the critical point is stable or unstable depending on the sign of $f'(x_0)$. Explain. Hint: see the “Autonomous equations” section.