The Extreme Value Theorem and Local Extreme Value Theorem

#calc/theorem

Very similar to the Intermediate Value Theorem.

Extreme value theorem

If $f$ is continuous on a closed interval $[a,b]$, then $f$ has both a maximum and minimum value on the interval.

Very simple.

Local extreme value theorem

If a function $f$ has a local maximum value or a local minimum value at an interior point $c$ of its domain, and if $f^{\prime }$ exists at $c$, then $f^{\prime }(c)=0$.

• If there’s a local minimum or maximum, then the rate of change will be $0$ at that point.