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Comparing Derivatives

  1. Sep 19, 2013 #1
    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. jcsd
  3. Sep 19, 2013 #2

    tiny-tim

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    Hi Pranav-Arora! :smile:

    Hint: consider g - 4f and 3g - 4f :wink:
     
  4. Sep 19, 2013 #3
    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?
     
  5. Sep 19, 2013 #4

    tiny-tim

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    yup! :smile:

    and h' = g' - 4f' :wink:
     
  6. Sep 19, 2013 #5
    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:
     
  7. Sep 19, 2013 #6

    tiny-tim

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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:
     
  8. Sep 19, 2013 #7
    Okay, I understand, thank you once again! :)
     
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