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Rolle's theorem -> Differentiability

  1. Mar 29, 2013 #1
    1. The problem statement, all variables and given/known data
    So I'm doing problems where I have to verify Rolle's hypotheses. I am only having trouble with the differentiability part. My professor wants me to prove this. So for example,
    f(x)=√(x)-(1/3)x [0,9]

    2. Relevant equations
    none

    3. The attempt at a solution
    1.) I know the function is continuous because root functions are continuous on their domains and polynomials are continuous everywhere therefore the difference of two continuous functions is continuous.

    2.) What could I say about it being differentiable?
    My GUESS: it is defined everywhere on its domain [0,9] and continuous it is therefore differentiable on its domain [0,9]?

    I am only stuck on 2.) so no need to go further on the problem.
     
  2. jcsd
  3. Mar 29, 2013 #2

    HallsofIvy

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    No. Saying that a function is "defined and continuous" on a domain does NOT mean that it is differentiable on the interval. For example, |x| is "defined and continuous" for all x but is not differentiable at x= 0.

    It's hard to tell you how you should answer this without knowing what you about derivatives. IF you know that the derivative of [itex]x^n[/itex] is [itex]nx^{n-1}[/itex] then it should be easy to tell what the derivative of [itex]x^{1/2}- x/3[/itex] is and so where it is differentiable.

    (It is NOT differentiable on [0, 1] but Rolle's theorem does not require it to be.)
     
  4. Mar 29, 2013 #3
    I know the derivative to the function. I also know chain rule and the limit definition as well quotient rule, etc. So the way my professor explained is that I have to verify I can differentiate over the interval given. In this case [0,9].

    So I know f'(x)=1/(2√x)-(1/3)

    I just don't understand how I verify that is differentiable over [0,9].

    Here is how he gave me Rolle's Theorem (very simply):
    1.) f is continuous on [a,b] (Prove it!)
    2.) f is differentiable on (a,b) (Prove it!)
    3.) f(a)=f(b) then if all 3 conditions are met a<c<b such that c belongs to (a,b) and that f'(c)=0.

    I hope this helps you understand my situation. I am only stuck on 2.).... The rest is I understand.

    Also, thank you for replying. I appreciate all the help I can get.
     
  5. Mar 29, 2013 #4

    jbunniii

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    Well, you have computed a formula for the derivative. Are there any values of ##x## in ##[0,9]## that cause problems?
     
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