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if i wanted to graph
f prime of x and i'm given f x now
sometimes guys this is helpful sometimes
you get confused and everything's like
you're trying to figure everything out
and like sometimes it's easier and or
better just to kind of like sketch the
graph to kind of wrap your head around
everything so let's pretend everything
is
let's look at
f prime x now remember f prime of x is
talking about the
slopes of f of x correct
yes
okay so
if we're going to talk about f prime of
x if we're talking about um
values of f prime of x or less than zero
that means the slope of f of x is less
than zero so look at these slopes
negative negative negative right
and then it changes from negative to
positive here at three
so that means that's going to be a
critical value right where f of x is
equal to zero that's a critical value
that's a relative extrema
then it also changes at two
and also changes at five
it changes from a negative to a positive
okay so
now again we have no idea what the graph
looks like we're just estimating we're
sketching
so
from negative 7 to 3 we have a negative
slope right
so that means my f prime of x needs to
be
negative it needs to be less needs to be
negative
so we're changing from there
all right
then
now it's positive from negative three to
two
yes
did you guys agree
no don't agree
so it's gonna be positive
from
there and then from two to five you can
see the slopes are always negative right
so from two to five we're going to be
dealing with a negative
surprising
and then we can just draw something like
that
okay so the one thing i want you guys to
notice when you're going from f to f
prime notice how we kind of lost a
turning point
right
we lost a turning point and again that
kind of makes sense like if you had
you know f of x equaled x to the fourth
and then we do f prime of x well now
we're looking at four x cubed right so
we're losing a curve we're losing the
turning points and then if we did f
double prime
we're looking at um 12x squared
right so now we want to do is look at
when are the slopes changing
so we look at the slopes changing guys
and we see that the slopes of f double
prime of x
so we'll do blue is f double prime of x
the slopes are changing
right here
and
there
yes also f double prime of x also tells
us about
concavity and concavity
occurs at the
points of inflection where would the
second derivative would be equal to zero
so therefore
these since it's changing here
these are going to be my new x
intercepts
for my second derivative
now look at these points on the f of x
graph is that where the graph changes
concavity
does the black graph change concavity at
these blue dots
it does
it does
right so when the second derivative is
equal to zero that's the points of
inflection that's also where the
concavity changes of f of x
now let's look at the derivative here
from here to here or the slope here here
is the slope positive or negative of f
prime of x
what are the slopes positive negative
zero it's all positive so my graph
should be opening up like that something
then from here to here the slope is
negative
then from here to here it's going to be
going back up
positive
okay
guys
this is not on your
quiz we are going to spend some more
time
