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Homework Help: Finding f inverse prime at a point c

  1. Nov 3, 2012 #1
    1. The problem statement, all variables and given/known data
    Assume the function f defined by f(x)=5x+sin(πx) is strictly increasing on ℝ. Find (f[itex]^{-1}[/itex])'(10)


    2. Relevant equations
    Let I and J be be intervals and let f:I->J be a continuous, strictly monotone function. If f is differentiable at c and if f'(c)≠0, then (f[itex]^{-1}[/itex]) is differentiable at f(c) and (f[itex]^{-1}[/itex])'(f(c))= 1/f'(c)


    3. The attempt at a solution

    It is clear f is continuous and differentiable on ℝ.
    => f'(x) = 5+πcos(πx)


    Finding when f(x)=10,
    10 = 5x+sin(πx) => x=2

    Then (f[itex]^{-1}[/itex])'(f(2))=1/f'(2) = 1/(5+πcos(2π)) = 1/(5+(π))

    Is this how to do it, or do I use f(10) instead of finding when f(x) is 10?
     
  2. jcsd
  3. Nov 4, 2012 #2

    haruspex

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    That all looks right. If you think of it as y = f(x) and x = f-1(y), the 10 is a value of y, not of x, so f(10) and f'(10) would not be relevant.
     
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