Question

Help with Increasing and Decreasing Derivatives Please!!

Answer 1

what is the question exactly?

Answer 2

If f'>0, then f is increasing
If f'<0, then f is decreasing

Answer 3

Oh...sorry I thought I had the attachment on

Answer 4

I neeed help with C and D

Answer 5

@myininaya

Answer 6

so are you having problems with the part highlighted in yellow?

Answer 7

Basically...

Answer 8

ok then so you know that g' will tell us if g is decreasing or increasing right?

Answer 9

Yeah

Answer 10

so find g' given that:

\[g(x)=\int\limits_{0}^{x} f(t) dt \]
don't look at the picture yet
just tell what g' is

Answer 11

f(x)+0.5

Answer 12

F(x)+c right?

Answer 13

no i'm sorry that isn't correct

ok let's look at this...
I will rewrite it a little...
\[g(x)=F(x)-F(0)\]
where F'=f

Now differentiate to find g'

Answer 14

I'm sorry..so what is g'?

Answer 15

you have to differentiate g(x)=F(x)-F(0) to find g'
can you do that?

Answer 16

I'm not sure what differentiate means anymore?

Answer 17

to find derivative

Answer 18

It's f'(x)-f'(0)

Answer 19

f'(x)(x) right?

Answer 20

I will help you out some more.
derivative of g is g'
derivative of F is f (this was given above when I said F'=f)
derivative of a constant is 0

everything i said in this little post right here will need to be used

Answer 21

oh....so the derivative of F(x) is f(x). Is that what you mean?

Answer 22

That is what I said

Answer 23

so do you know F(0) is a constant?

Answer 24

its 0.5

Answer 25

I think you are thinking of f(0) not F(0)
f is given not F

Answer 26

yea?

Answer 27

Anyways F(0) is a constant.
Because F is a just a function of x
any if you plug in a number for x then you will definitely receive a constant





----------------------------------------------------
example:


Say F(x)=cos(x)
well F(0) is definitely a constant because F(0) is 1 and 1 never changes (it is and will always remain 1)

Answer 28

So going back to \[g(x)=\int\limits_{0}^{x} f(t) dt \\ g(x)=F(x)-F(0)\]
can you differentiate g now?

Answer 29

what that means is you will have to differentiate both sides (not just one side)

Answer 30

Can you just tell me where it is increasing and concave up...so I'll try to figure it out?

Answer 31

Try to use what I said earlier...
derivative of g is g'
derivative of F' is f
derivative of a constant is 0

you can do this

Answer 32

I would prefer the answer because I have to write the explanation
anyway.

Answer 33

Okay...so you g(x)= F(x)-F(0)

Answer 34

that means g'(x) = f(x)-0?

Answer 35

I'm not going to give just the answer. Sorry.
But right g'=f
so that means the picture given is g'

Answer 36

and you know if g'>0, then g is increasing
and you know if g'<0, then g is decreasing

Answer 37

where on the picture that is given is g' above the x axis (because that is where g is increasing)


and when g' is below the x-axis that is where g is decreasing

Answer 38

what about concave up?