2.5 Critical Numbers
Today in class we learned about critical numbers. The
squares/rectangles indicate the local maximums and
the stars indicate the relative maximums.
It is called local because there
is no maximum or minimum but
relatively there are n number
of maximums and minimums.
In this case there are three
maximums and three
A local/relative maximum for
a function of f occurs at a
point x=a, if f(a) is the largest
value of f in some interval
centered at x=a
local/relative minimum >
Next, we learned how to draw f'(x).
where f is decreasing on
where f is increasing on
(a,b), f'(x) > 0
at the local extreme
A number C in the interior of the domain of a function f, is
called a critical number if either:
f'(c)=0 or f'(c)= d.n.e [does not exist]
local minimum of f local maximum of f
For this example we found the local extreme values by substituting 0 in for
dy/dx. We found two values 0 and 2 to be the critical points.
We then used a number line to find out where negative and positive outcomes
From our information, we were able to draw the resulting graph.
*Note* All critical points are not local extreme values.
This is not a local extremum. It
is called a point of inflection .
This is an example of where the critical point is not a local extreme value.
Again, we solved for our critcal point by substituting zero into the equation
and used a number line to find negative and/or positive outcomes.
Then we drew the graphs of f'(x) and f(x).