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Math Help - differentiability and continuity question

  1. #1
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    differentiability and continuity question

    A function f(x) is defined to equal cos(ax) + b for x>=0 and 2 x3 for x < 0, where a and b are constants. It is known that this function is differentiable everywhere. Find all possible values of a and b

    My answer was that a could be anything and b has to have a value of 2. Could someone confirm this. Thank you.
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  2. #2
    MHF Contributor Mathstud28's Avatar
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    Quote Originally Posted by geometry101 View Post
    A function f(x) is defined to equal cos(ax) + b for x>=0 and 2 x3 for x < 0, where a and b are constants. It is known that this function is differentiable everywhere. Find all possible values of a and b

    My answer was that a could be anything and b has to have a value of 2. Could someone confirm this. Thank you.
    So we have

    f(x)=\left\{ \begin{array}{ll} -x^3+2 & \mbox{ if } {x<0}\\ \cos(ax)+b & \mbox{ if }{x\geq{0}}\end{array}\right.

    Now for it to be differentiable we must have that

    \lim_{x\to{0^{-}}}f(x)=\lim_{x\to{0^+}}f(x)

    and \lim_{x\to{0^-}}f'(x)=\lim_{x\to{0^+}}f'(x)

    Now can you see what to do?
    Last edited by Mathstud28; June 15th 2008 at 12:50 PM. Reason: Forgot the or equal to
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