The Extreme Value Theorem tells us that the minimum and maximum
of a continuous function on a closed interval have to be
somewhere. But where should we look?

It turns out that extreme values can occur at the endpoints of
the intervals, or at the "top of a hill" or the "bottom of a
valley" of the graph of a function. How can we find these
hills and valleys? They can occur only at $x$-values where
the derivative either equals zero (where
the tangent line has slope of zero), or does not exist (where
there may be a cusp).

Fermat's
Theorem: Suppose that $a \lt c \lt b$. If a function
$f$ is defined on the interval $(a,b)$, and it has a maximum
or a minimum at $c$, then either $f'$ doesn't exist at $c$
or $f'(c)=0$.

Such $x$-values $c$ are called critical
numbers.

Critical
numbers: A critical number of a function $f$ is
a number $c$ in the domain of $f$ such that $f'(c)=0$ or
$f'(c)$ is not defined.

What we are saying, then, is that absolute
maxima or minima can
only occur at critical numbers and endpoints of intervals.

Equivalently, if $f'(c)$ exists and is not
zero, then $f(c)$ is neither a maximum nor a minimum.

The process of finding such maxima and minima is not difficult in
practice: