# What are local extrema?

Dec 13, 2016

Points on some function where a local maximum or minimum value occurs. For a continuous function over its entire domain, these points exist where the slope of the function $= 0$ (i.e it's first derivative is equal to 0).
Consider some continuous function $f \left(x\right)$
The slope of $f \left(x\right)$ is equal to zero where $f ' \left(x\right) = 0$ at some point $\left(a , f \left(a\right)\right)$. Then $f \left(a\right)$ will be a local extreme value (maximim or minimum) of $f \left(x\right)$
N.B. Absolute extrema are a subset of local extrema. These are the points where $f \left(a\right)$ is the extreme value of $f \left(x\right)$ over its entire domain.