Difficult Question in Calculus — limits and integrals

[omeraz100]

Homework Statement

(hebrew) : f(x) a continuous function. proof the following

Homework Equations

I guess rules of limits and integrals

The Attempt at a Solution

I've tried several approaches:
taking ln() of both sides and using L'Hospitale Rule.
Thought about using integral reduction formula.
But really nothing even got me close.
Looking forward to any advice

 

[Mark44]

Homework Statement

(hebrew) : f(x) a continuous function. proof the following

Homework Equations

I guess rules of limits and integrals

The Attempt at a Solution

I've tried several approaches:
taking ln() of both sides and using L'Hospitale Rule.
Thought about using integral reduction formula.
But really nothing even got me close.
Looking forward to any advice

Please show us at least some of what you've tried. Per forum rules, you must show an attempt.

 

[omeraz100]

wrong link
here again:
<a href=http://www.filedropper.com/2015-03-2518-01-01><img src=http://www.filedropper.com/download_button.png width=127 height=145 border=0/></a><br /><div style=font-size:9px;font-family:Arial, Helvetica, sans-serif;width:127px;font-color:#44a854;> <a href=http://www.filedropper.com >file storage online</a></div>

 

[Dick]

wrong link
here again:
<a href=http://www.filedropper.com/2015-03-2518-01-01><img src=http://www.filedropper.com/download_button.png width=127 height=145 border=0/></a><br /><div style=font-size:9px;font-family:Arial, Helvetica, sans-serif;width:127px;font-color:#44a854;> <a href=http://www.filedropper.com >file storage online</a></div>

Here's a hint. Suppose f is a step function http://en.wikipedia.org/wiki/Step_function. Can you prove it in that case?

 

[Ray Vickson]

Homework Statement

(hebrew) : f(x) a continuous function. proof the following

Homework Equations

I guess rules of limits and integrals

The Attempt at a Solution

I've tried several approaches:
taking ln() of both sides and using L'Hospitale Rule.
Thought about using integral reduction formula.
But really nothing even got me close.
Looking forward to any advice

If I were doing it would let
$$M = \max_{x \in [a,b]} |f(x)|, \; L_t = \left( \int_a^b |f(x)|^t \, dx \right)^{1/t},$$
then I would show that: (1) $\lim_{t \to \infty} L_t \leq M$; and (2) $\lim_{t \to \infty} L_t \geq M$.

(1) is quite easy; (2) is a bit trickier and needs some properties of continuous functions on finite, closed intervals.