# How do I find the extrema of a function?

May 3, 2018

Check below.

#### Explanation:

Given a point $M \left({x}_{0} , f \left({x}_{0}\right)\right)$, if $f$ is decreasing in $\left[a , {x}_{0}\right]$ and increasing in $\left[{x}_{0} , b\right]$ then we say $f$ has a local minimum at ${x}_{0}$, $f \left({x}_{0}\right) = \ldots$

If $f$ is increasing in $\left[a , {x}_{0}\right]$ and decreasing in $\left[{x}_{0} , b\right]$ then we say $f$ has a local maximum at ${x}_{0}$, $f \left({x}_{0}\right) = \ldots .$

More specifically, given $f$ with domain $A$ we say that $f$ has a local maximum at ${x}_{0}$$\in$$A$ when there is δ>0 for which
$f \left(x\right) \le f \left({x}_{0}\right)$ , $x$$\in A \cap$(x_0-δ,x_0+δ) ,
In similar way, local min when $f \left(x\right) \ge f \left({x}_{0}\right)$
If $f \left(x\right) \le f \left({x}_{0}\right)$ or $f \left(x\right) \ge f \left({x}_{0}\right)$ is true for ALL $x$$\in$$A$ then $f$ has an extrema (absolute)

If $f$ has no other local extremas in its domain ${D}_{f}$ then we say $f$ has an extrema (absolute) at ${x}_{0}$.

Creating a monotony table in each case where you can study $f '$ sign and $f$ monotony in their domain will make things easier.