# help with this integral problem

• May 21st 2008, 07:44 PM
szpengchao
help with this integral problem
• May 21st 2008, 09:56 PM
kalagota
for the first one, it just follows directly from $m \leq f(x) \leq M$ by taking the definite integrals in all parts (since integral preserves the direction of the inequality). Also note that constants can go out of the integral sign.

for the second part, since $g(x) \geq 0$, the inequality does not change i.e. $mg(x) \leq f(x)g(x) \leq Mg(x)$ and you can take the integrals.

the next two are applications.. you can do it.
• May 21st 2008, 11:09 PM
Isomorphism
Quote:
First apply parts to get,

$\int_{0}^{\frac1{\sqrt{n}}} nf(x) e^{-nx} \, dx = -f(x) e^{-nx} \bigg{|}_{0}^{\frac1{\sqrt{n}}} + \int_{0}^{\frac1{\sqrt{n}}} f'(x) e^{-nx} \, dx$

$\lim_{n \to \infty} -f(x) e^{-nx} \bigg{|}_{0}^{\frac1{\sqrt{n}}}$

$= \lim_{n \to \infty} -f(\frac1{\sqrt{n}}) e^{-\sqrt{n} x} + f(0) = f(0)$

[That is because $\lim_{n \to \infty} -f(\frac1{\sqrt n})e^{-\sqrt{n}x} = 0$ since $e^{-\sqrt{n}x} \to 0$ as $n \to \infty$]

We claim $\int_{0}^{\frac1{\sqrt{n}}} f'(x) e^{-nx} \, dx = 0$

First observe that $0 < e^{-nx} < 1$

So $0 < \int_{0}^{\frac1{\sqrt{n}}} e^{-nx}f'(x)\, dx < \int_{0}^{\frac1{\sqrt{n}}} f'(x) \, dx$

$0 < \lim_{n \to \infty} \int_{0}^{\frac1{\sqrt{n}}} e^{-nx}f'(x)\, dx < \lim_{n \to \infty} f\left(\frac1{\sqrt{n}}\right) - f(0) = f(0)\text{(Why?) }- f(0) = 0$

Thus claim proved by sandwich theorem...

So
$\lim_{n \to \infty} \int_{0}^{\frac1{\sqrt{n}}} nf(x) e^{-nx} \, dx = f(0)$

EDIT: I realised this halfway through...(Worried) My proof is incomplete... I assumed (1) f is differentiable (2) f'(x) > 0.... Let me think about it a little more....
• May 22nd 2008, 05:29 AM
kalagota
as i have said, the questions are just applications of the previous.

note that, $f$ is continuous in a closed and bounded interval, then you can use Maximum-Minimum theorem, that is $f$ is also bounded. Also, $f$ is integrable.

Let $m, M$ be real numbers such that $m \leq f(x) \leq M$ for $x \in [0,1]$. Applying the first and second facts, we have $m \int_0^{1/\sqrt{n}} n e^{-nx}dx \leq \int_0^{1/\sqrt{n}} nf(x)e^{-nx}dx \leq M \int_0^{1/\sqrt{n}} n e^{-nx}dx$ (since $1/\sqrt{n} \in [0,1]$)

i just wonder if it equal $f(0)$ or $0$ unless $f(0) = 0$
• May 22nd 2008, 12:44 PM
Isomorphism
Quote:

Originally Posted by kalagota
as i have said, the questions are just applications of the previous.

note that, $f$ is continuous in a closed and bounded interval, then you can use Maximum-Minimum theorem, that is $f$ is also bounded. Also, $f$ is integrable.

Let $m, M$ be real numbers such that $m \leq f(x) \leq M$ for $x \in [0,1]$. Applying the first and second facts, we have $m \int_0^{1/\sqrt{n}} n e^{-nx}dx \leq \int_0^{1/\sqrt{n}} nf(x)e^{-nx}dx \leq M \int_0^{1/\sqrt{n}} n e^{-nx}dx$ (since $1/\sqrt{n} \in [0,1]$)

i just wonder if it equal $f(0)$ or $0$ unless $f(0) = 0$

That does not lead us anywhere, but tell us that the integral is between M and m!