# Homework Help: How he concluded that?Spivak's calculus

1. Aug 9, 2013

### Andrax

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?

$2. Relevant equations included above 1. The problem statement, all variables and given/known data 2. Aug 9, 2013 ### ehild f(x)=0 is also a function, and solution of the problem. But there is also an other "non-trivial" solution. ehild 3. Aug 9, 2013 ### Andrax so when he wrotes whenever f(x)=/=0 he mean't the function f(x)=0 not the points where f(x)=0?it makes sense now 4. Aug 10, 2013 ### ehild Yes, he wants to find the solution not identically zero. ehild 5. Aug 10, 2013 ### LCKurtz Does that mean$(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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