Results 1 to 4 of 4

Thread: differentiability

  1. #1
    Junior Member
    Joined
    Dec 2011
    Posts
    57
    Thanks
    1

    differentiability

    Suppose $\displaystyle g:\mathbb{R}\rightarrow{\mathbb{R}}$ is a twice differentiable function with $\displaystyle g(0)=g'(0)=0$ and $\displaystyle g''(0)=14$. Let $\displaystyle f:\mathbb{R}\rightarrow{\mathbb{R}}$ be defined by
    $\displaystyle f(x)=\frac{g(x)}{x}$ if $\displaystyle x\neq{0}$, $\displaystyle 0$ if $\displaystyle x=0$
    Prove that $\displaystyle f$ is differentiable at $\displaystyle x=0$ and find $\displaystyle f'(0)$.


    Can we use L' Hopital's Rule to solve this question?
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Junior Member
    Joined
    Jun 2011
    Posts
    34

    Re: differentiability

    Use L' Hopital's Rule to find the limit of f(x) at x=0 first.

    f(x) = g(x)/x, since g is a differentiable function,
    therefore f(x) must be differentiable somewhere on R.
    assume f'(x) exists at point x=k, then

    f'(k) = [k g'(k) - g(k) ]/k^2. (by quotient rule)
    take above expression become a limit which k approach 0 and apply L' Hopital's Rule to show that f is differentiable at x=0, and hence find f'(0), the answer should be 7.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Junior Member
    Joined
    Dec 2011
    Posts
    57
    Thanks
    1

    Re: differentiability

    Quote Originally Posted by piscoau View Post
    Use L' Hopital's Rule to find the limit of f(x) at x=0 first.

    f(x) = g(x)/x, since g is a differentiable function,
    therefore f(x) must be differentiable somewhere on R.
    assume f'(x) exists at point x=k, then

    f'(k) = [k g'(k) - g(k) ]/k^2. (by quotient rule)
    take above expression become a limit which k approach 0 and apply L' Hopital's Rule to show that f is differentiable at x=0, and hence find f'(0), the answer should be 7.
    $\displaystyle \lim_{k\to0}f'(k)=\lim_{k\to0}\frac{kg'(k)-g(k)}{k^2}=\lim_{k\to0}\frac{kg''(k)+g'(k)-g'(k)}{2k}=\lim_{k\to0}\frac{g''(k)}{2}=7$

    Is this correct?
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Junior Member
    Joined
    Jun 2011
    Posts
    34

    Re: differentiability

    YES
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. differentiability
    Posted in the Pre-Calculus Forum
    Replies: 3
    Last Post: May 2nd 2011, 03:47 PM
  2. differentiability at x=0
    Posted in the Calculus Forum
    Replies: 1
    Last Post: Jan 8th 2011, 08:39 PM
  3. Differentiability
    Posted in the Calculus Forum
    Replies: 3
    Last Post: Oct 13th 2010, 12:45 PM
  4. Differentiability
    Posted in the Calculus Forum
    Replies: 2
    Last Post: Sep 21st 2010, 08:16 PM
  5. Differentiability
    Posted in the Calculus Forum
    Replies: 1
    Last Post: Oct 20th 2009, 12:14 PM

Search Tags


/mathhelpforum @mathhelpforum