Help with Increasing and Decreasing Derivatives Please!!

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- SanjanaP

Help with Increasing and Decreasing Derivatives Please!!

- chestercat

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- myininaya

what is the question exactly?

- myininaya

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

- SanjanaP

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

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- SanjanaP

I neeed help with C and D

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- SanjanaP

- myininaya

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

- SanjanaP

Basically...

- myininaya

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

- SanjanaP

Yeah

- myininaya

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

- SanjanaP

f(x)+0.5

- SanjanaP

F(x)+c right?

- myininaya

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'

- SanjanaP

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

- myininaya

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

- SanjanaP

I'm not sure what differentiate means anymore?

- myininaya

to find derivative

- SanjanaP

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

- SanjanaP

f'(x)(x) right?

- myininaya

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

- SanjanaP

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

- myininaya

That is what I said

- myininaya

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

- SanjanaP

its 0.5

- myininaya

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

- SanjanaP

yea?

- myininaya

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)

- myininaya

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

- myininaya

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

- SanjanaP

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

- myininaya

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

- SanjanaP

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

- SanjanaP

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

- SanjanaP

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

- myininaya

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

- myininaya

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

- myininaya

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

- SanjanaP

what about concave up?

- SanjanaP

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