Math 22 Extrema and First Derivative Test

Relative Extrema

 Let $f$ be a function defined at $c$.
 1. $f(c)$ is a relative maximum of $f$ when there exists an interval $(a,b)$ containing $c$ such that $f(x)\leq f(c)$ for all $x$ in $(a,b)$.
 2. $f(c)$ is a relative minimum of $f$ when there exists an interval $(a,b)$ containing $c$ such that $f(x)\geq f(c)$ for all $x$ in $(a,b)$.

If $f$ has a relative minimum or relative maximum at $x=c$, then $c$ is a critical number of $f$. That is, either $f'(c)=0$ or $f'(c)$ is undefined.

Relative extrema must occur at critical numbers as shown in picture below.

The First-Derivative Test

 Let $f$ be continuous on the interval $(a,b)$ in which $c$ is the only critical number, then
 
 On the interval $(a,b)$, if $f'(x)$ is negative to the left of $x=c$ and positive to the right of $x=c$, then $f(c)$ is a relative minimum.
 
 On the interval $(a,b)$, if $f'(x)$ is positive to the left of $x=c$ and negative to the right of $x=c$, then Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(c)}
 is a relative maximum.

Absolute Extrema

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