Critical point

Let $f:I\to\mathbb{R}$ be defined on an interval $I\subseteq\mathbb{R}$, and let $a\in I$. The point $a$ is a critical point of $f$ if either:

• $f'(a)=0$, or
• $f'(a)$ does not exist.

Critical points are where local extrema can occur (Fermat’s principle: interior local extrema force $f'(a)=0$ when differentiable ). They are the main inputs for optimization via derivative tests.

Examples:

• For $f(x)=x^2$, $a=0$ is a critical point since $f'(0)=0$.
• For $f(x)=|x|$, $a=0$ is a critical point since $f'(0)$ does not exist.
• For $f(x)=e^x$, there are no critical points since $f'(x)=e^x\ne 0$ everywhere.