Comparing Derivatives

1. Sep 19, 2013

Saitama

1. The problem statement, all variables and given/known data
Let $f(x)$ and $g(x)$ be two differentiable function in R and f(2)=8, g(2)=0, f(4)=10 and g(4)=8 then

A)$g'(x)>4f'(x) \forall \, x \, \in (2,4)$

B)$3g'(x)=4f'(x) \, \text{for at least one} \, x \, \in (2,4)$

C)$g(x)>f(x) \forall \, x \, \in (2,4)$

D)$g'(x)=4f'(x) \, \text{for at least one} \, x \, \in (2,4)$

2. Relevant equations

3. The attempt at a solution
How am I to compare the derivatives with only two points? I really don't know where to start with this. Just a wild guess, do I need to apply the mean value theorem?

2. Sep 19, 2013

tiny-tim

Hi Pranav-Arora!

Hint: consider g - 4f and 3g - 4f

3. Sep 19, 2013

Saitama

I don't see how does it help.

Let $h(x)=g(x)-4f(x)$. Then $h(4)=h(2)=-32$. This suggests that h' is zero somewhere in (2,4). What should I do now?

4. Sep 19, 2013

tiny-tim

yup!

and h' = g' - 4f'

5. Sep 19, 2013

Saitama

Ah yes, then its D. Thank you tiny-tim!

Let v(x)=3g(x)-4f(x). How do I prove that v'(x) is never zero in (2,4)?

6. Sep 19, 2013

tiny-tim

B is just there to confuse you!

Anyway, you'd only need to show that it can be never-zero …

you should be able to sketch a counter-example.

7. Sep 19, 2013

Saitama

Okay, I understand, thank you once again! :)