4.1 - Extreme Values of Functions

What are absolute extrema?

If $f$ is a function with domain $D$, then $f(c)$ is the absolute maximum value on $D$ if and only if $f(x)\le f(c)$ for all $x$ in $D$.

• Essentially, the absolute maximum value is only an absolute maximum if it’s bigger than everything else.
  If $f$ is a function with domain $D$, then $f(c)$ is the absolute minimum value on $D$ if and only if $f(x)\ge f(c)$ for all $x$ in $D$.
• Same thing but the opposite way for the absolute minimum value.

The Extreme Value Theorem

What are local extrema?

The same as above, but in some interval.

• Keep in mind that absolute extrema also count as local extrema.

The Local Extreme Value Theorem

Key info

• Extreme values can occur at critical points OR endpoints.
• Using a sign chart on a derivative is essential, because a function’s derivative does not always change sign at critical points.