The First Derivative Test
Remember that critical numbers are the only places where a function
can have a local maximum or minimum, and are the only places where
$f'(x)$ can change sign. These ideas are related by
The First Derivative Test: Let
$c$ be a critical number for a continuous function $f$
- If $f'(x)$ changes from positive to negative at $c$,
then $f(c)$ is a local maximum.
- If $f'(x)$ changes from negative to positive at $c$,
then $f(c)$ is a local minimum.
- If $f'(x)$ does not change sign at $c$, then $f(c)$ is
neither a local maximum or minimum.
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Notice: A critical number may indicate a local extrema,
but some critical numbers do not
yield local extreme values. This is the
circumstance indicated in number 3.
above.
In the examples below, we have included both $f$ and $f'$ in
each of the 8 graphs. You can see the relationship between
the sign of $f'(x)$ (the slope of the tangent line at $x$,
positive or negative) and the direction of $f$ at $x$
(increasing or decreasing). DO:
Study these examples. Find those where the critical
number does not yield a max or min.
The first derivative test makes sense. A local maximum is where
you stop going up and start coming down. A local minimum is where
you stop going down and start coming up. If you flatten out and
then resume going in the direction you were already heading,
you're at a critical point but not at a maximum or minimum. Once
you make a number line with intervals between critical numbers for
$f'(x)$, as we outlined previously, then you will know where all
the local extreme values are.
Example: Find the local maxima and minima of $f(x)=x^3-3x^2$.
Solution: In a previous slide, we determined that
$f'(x)$ is positive on $(-\infty,0)$, negative on $(0,2)$ and
positive on $(2,\infty)$. Since $f'$ goes from positive to negative
at $x=0$, there is a local maximum at $x=0$. Since $f'$ goes from
negative to positive at $x=2$, there is a local minimum at $x=2$.
