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I've got a exercise about differentiability, mean value theorem and suprema.

To be honest I don't understand the structure of this question. Maybe you guys are so kind to help me out :)

Let $f: [0,1] \rightarrow \mathbb{R}$ be differentiable with $f(0) = 0$, and satisfying

$$|f'(x)|\le M|f(x)|, x\in[0,1] $$ for some $M>0$

a.) Use the Mean Value Theorem to show that for all $x \le x_0 \in[0,1], y\in[0,x_0]:$ $$ |f(x)|\le x_0 \text{ sup} |f'(y)|\le M x_0 \text{ sup} |f(y)|$$ b.) Use the previous part to show that f is the zero-function on [0,1].

(Hint: What happens if we choose $x_0$ such that M$x_0<1$?)

Known definitions, theorems:

  • Mean Value Theorem (There is a c for which f'(c) equals f(b)-f(a)/(b-a) if [a,b] is the domain, and f is continuous on [a,b], differentiable on (a,b)
  • Differentiable function means that that for all points c $\in$ A, the limit of f(x)-f(c)/(x-c) exists.
  • Interior Extremum Theorem (Intermediate Value theorem). States that if f attains a maximum or minimum value on a open interval, then at some point c $\in$(a,b), f'(c)=0
  • Darboux Theorem: If f is differentiale on an interval [a,b] and if if $\alpha$ satistfies $f'(a) < \alpha < f'(b)$, then there exists a point c$\in$(a,b) where $f(c)=\alpha$
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Nobody? Please help me out, because I don't understand even the question :( –  MSKfdaswplwq Oct 23 '12 at 0:38
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1 Answer

up vote 2 down vote accepted

Let f:[0,1]→R be differentiable with f(0)=0, and satisfying

$$|f'(x)|\le M|f(x)|, x\in[0,1] $$ for some $M>0$

a.) Use the Mean Value Theorem to show that for all $x \le x_0 \in[0,1], y\in[0,x_0]:$ $$ |f(x)|\le x_0 \sup |f'(y)|\le M x_0 \sup |f(y)|$$ b.) Use the previous part to show that f is the zero-function on [0,1].

Let's begin with part a. So fix an $x_0$ in $(0,1)$. Suppose that there is an $x$ in $[0,x_0]$ such that $|f(x)| > x_0 \sup|f'(y)|$. Then in particular, $\dfrac{|f(x) - f(0)|}{x-0} = \dfrac{|f(x)|}{x} > \dfrac{x_0}{x}\sup f'(y) \geq \sup f'(y)$ as $x_0 \geq x$.

But then by the mean value theorem, there must exist a $c$ such that $f'(c) = \dfrac{f(x) - f(0)}{x - 0}$. This is a contradiction, as then $f'(c) > \sup f'(y)$.

The second inequality is much easier, relying just on using $|f(x)|\le x_0 \sup |f'(y)|\le M x_0 \sup |f(y)|$ and interpreting sup.

And then you can follow the hint for part b, and the answer falls out as $|f(x)| < \sup |f(y)|$ is nonsense.

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you are great, thank you so much :) –  MSKfdaswplwq Oct 23 '12 at 11:09
    
can you explain why it is not possible that $f'(c)> \sup f'(y)$ ? And you forgot to write all the absolute value symbols right? –  MSKfdaswplwq Oct 23 '12 at 12:25
    
@Hempo: Ah, I did forget a few absolute values. There's more or less one around everything. And $f'(c)$ can't be greater than $\sup f'(y)$ because it's the sup, and so is at least as big as any other. –  mixedmath Oct 23 '12 at 14:36
    
@mixedmath A small question. If we pick $x_0$ such that $x_0M < 1$ then the supremum is taken over $y\in[0,\ x_0]$. Certainly the function has to be zero over that interval, but I'm a bit confused as to whether this argument shows the function is $0$ over $(x_0,\ 1]$. –  EuYu Oct 23 '12 at 15:47
    
@Euyu: I've only written down the proof for $[0,x_0]$, but it's easy to extend this. In particular, the key idea is that we've shown $f(x_0) = 0$ now. So if we apply the MVT again, we can see that $|f(x + x_0) - f(x_0)| \leq x_0 M \sup |f(y)|$ too, and repeating by translation. –  mixedmath Oct 23 '12 at 18:38
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