Comparing Derivatives

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  • #1
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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?
 

Answers and Replies

  • #2
tiny-tim
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Hi Pranav-Arora! :smile:

Hint: consider g - 4f and 3g - 4f :wink:
 
  • #3
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Hi Pranav-Arora! :smile:

Hint: consider g - 4f and 3g - 4f :wink:
I don't see how does it help. :confused:

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
tiny-tim
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This suggests that h' is zero somewhere in (2,4).
yup! :smile:

and h' = g' - 4f' :wink:
 
  • #5
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yup! :smile:

and h' = g' - 4f' :wink:
Ah yes, then its D. Thank you tiny-tim! :smile:

Let v(x)=3g(x)-4f(x). How do I prove that v'(x) is never zero in (2,4)? :confused:
 
  • #6
tiny-tim
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ALet v(x)=3g(x)-4f(x). How do I prove that v'(x) is never zero in (2,4)? :confused:
B is just there to confuse you! :wink:

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

you should be able to sketch a counter-example. :smile:
 
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  • #7
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B is just there to confuse you! :wink:

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

you should be able to sketch a counter-example. :smile:
Okay, I understand, thank you once again! :)
 

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