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Math Help - Calculus Problem - Properties of a Function

  1. #1
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    Calculus Problem - Properties of a Function

    Hello

    This is my first post in this forum

    I have a mathematical problem that I could not solve. Could you please give me some hints how to solve it?

    Let f: [0,1] \rightarrow \mathbb{R} be a continous and on (0,1) a differentiable function with following properties:

    a) f(0) = 0
    b) there exists a M>0 with |f'(x)| \leq M |f(x)| for all x \in (0,1)

    Now the problem is: Show that f(x) = 0 is true for all x \in [0,1]

    There is a hint given but it doesn't help me The hint is: Consider the set D = \{ x \in [0,1]: ~ f(t) =0 for t \in [0,x] \} and show that the the supremum of this set is 1.

    Thanks for help
    Greetings
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  2. #2
    MHF Contributor

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    Re: Calculus Problem - Properties of a Function

    I think I would be inclined to use the "mean value" theorem. Suppose there exist x_1such that f(x_1)= y_1\ne 0. Then there exist x_2 between 0 and 1 such that f'(x_2)= \frac{f(x_1)- f(0)}{x_1- 0}= \frac{y_1}{x_1}
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