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How he concluded that?Spivak's calculus

  1. Aug 9, 2013 #1
    1. The problem statement, all variables and given/known data
    So the question is : Find all continious functions such that ##\displaystyle \int_{0}^{x} f(t) \, \mathrm{d}t##= ((f(x)^2)+C , what interests me is the way the solutions book presented the solution , not the solution itself .
    in the solution , it starts with this , clrealy f^2 is differentiable at every point ( it's derivative at x is f(x) ) So **f is differentiable at x whenever **f(x)=/=0****? I have no idea how that can be concluded , this is from Spivak's calculus , if you diffrentiate by the ftc it's clearly f(x)=f(x)f'(x)
    but he said that before even giving this formula, the differentiablity of f^2 dosen't have a relation with f(x)=/=0
    EDIT : I know that f(x)=f(x)f'(x) What i don't inderstand is this 'clrealy f^2 is differentiable at every point ( it's derivative is f) So f(x)=/=0" why f(x) mustn't equal 0?

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    2. Relevant equations
    included above
    1. The problem statement, all variables and given/known data
     
  2. jcsd
  3. Aug 9, 2013 #2

    ehild

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    f(x)=0 is also a function, and solution of the problem. But there is also an other "non-trivial" solution.

    ehild
     
  4. Aug 9, 2013 #3
    so when he wrotes whenever f(x)=/=0 he mean't the function f(x)=0 not the points where f(x)=0?it makes sense now
     
  5. Aug 10, 2013 #4

    ehild

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    Yes, he wants to find the solution not identically zero.

    ehild
     
  6. Aug 10, 2013 #5

    LCKurtz

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    Does that mean ##(f(x))^2## or ##f(x^2)##? I'm guessing the first.

    But ##f(x)f'(x)## is not the derivative of ##f^2(x)##. Am I missing something here?
     
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