Calculus Problem - Properties of a Function

**JoschuaS** · January 30th 2013, 08:31 AM

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

**HallsofIvy** · January 30th 2013, 08:53 AM

I think I would be inclined to use the "mean value" theorem. Suppose there exist $x_1$ such 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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