Comparing Derivatives: Solving for g'(x) and f'(x) with Two Given Points

[Saitama]

Homework Statement

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

Homework Equations

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?

 

[tiny-tim]

Hi Pranav-Arora!

Hint: consider g - 4f and 3g - 4f ​

 

[Saitama]

Hi Pranav-Arora!

Hint: consider g - 4f and 3g - 4f ​

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?

 

[tiny-tim]

This suggests that h' is zero somewhere in (2,4).

yup!

and h' = g' - 4f' ​

 

[Saitama]

yup!

and h' = g' - 4f' ​

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)?

 

[tiny-tim]

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

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. ​

 

[Saitama]

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. ​

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