112. Critical Points

A critical point of a function $𝑓\left(𝑥\right)$ is a point in the domain where either:

• ${𝑓}^{\prime }\left(𝑥\right)=0$

• ${𝑓}^{\prime }\left(𝑥\right)$ is undefined

112.0.1. Global vs. Local Extrema

• $𝑓\left(𝑐\right)$ is a relative maximum if $𝑓\left(𝑐\right)\ge 𝑓\left(𝑥\right)$ for all $𝑥\in \left(𝑐-ℎ,𝑐+ℎ\right)$ for $ℎ>0$

• $𝑓\left(𝑑\right)$ is a relative minimum if $𝑓\left(𝑑\right)\le 𝑓\left(𝑥\right)$ for all $𝑥\in \left(𝑑-ℎ,𝑑+ℎ\right)$ for $ℎ>0$

112.0.2. First and Second Derivative Tests

First Derivative Test and the Second Derivative Test are used to classify critical points, and both aim to determine whether the point is a local maximum, local minimum, or neither