# Comparing Derivatives

## 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)##

## 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
Homework Helper
Hi Pranav-Arora! Hint: consider g - 4f and 3g - 4f 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
Homework Helper
This suggests that h' is zero somewhere in (2,4).

yup! and h' = g' - 4f' 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
Homework Helper
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. • 1 person
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! :)