# How does the first derivative test work?

Let $x = c$ be a critical value of $f \left(x\right)$.
If $f ' \left(x\right)$ changes its sign from + to - around $x = c$, then $f \left(c\right)$ is a local maximum.
If $f ' \left(x\right)$ changes its sign from - to + around $x = c$, then $f \left(c\right)$ is a local minimum.
If $f ' \left(x\right)$ does not change its sign around $x = c$, then $f \left(c\right)$ is neither a local maximum nor a local minimum.