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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