Improper Integral help

[wassupman]

Can anyone point me in the right direction for this. How am I supposed to find f(x) from what they give me? I am confused on how to start this. Any help is appreciated.

 

[Steven G]

You have to try something.
If f(x) is defined on 2< x < infinity, then what values can x^8 + 10 be---give me ALL the values it can be. Then if |f(x)| < x^8 + 10, what values can f(x) take on?
Please post back with your work.

 

[skeeter]

integration by parts ...

[MATH]\int_3^\infty f(x) \cdot e^{-x/7} \, dx = \bigg[-7e^{-x/7} \cdot f(x) \bigg]_3^\infty + 7\int_3^\infty f'(x) \cdot e^{-x/7} \, dx[/MATH]
see what you can accomplish from here

 

[wassupman]

So here's my work so far. My teacher is saying that the f(3)=14 is useful in taking the limit. My question is, since |f(x)|<x^8+10 is a parabola going to positive infinity from 2<x<oo, how do I determine what f(x) is? All the values I get when plugging in values for x^8+10 are huge because its going to infinity. Thank you for the help.

 

[skeeter]

[MATH]-7e^{-x/7} \cdot f(x) = -\dfrac{7 \cdot f(x)}{e^{x/7}} \le -\dfrac{7(x^8+10)}{e^{x/7}}[/MATH]
so, what is the value of [MATH]\lim_{x \to \infty} -\dfrac{7(x^8+10)}{e^{x/7}}[/MATH] ... ?

 

[wassupman]

I think it would come to 0. Is that right? But if f(x) is zero then everything would be zero i think. sorry but Im still confused.

 

[skeeter]

I think it would come to 0. Is that right? But if f(x) is zero then everything would be zero i think. sorry but Im still confused.

f(x) isn’t zero, but the product, [MATH]f(x) \cdot (-7e^{-x/7}) \to 0 \text{ as } x \to \infty[/MATH]
[MATH]\bigg[-7e^{-x/7} \cdot f(x) \bigg]_3^\infty = -7\lim_{b \to \infty} \bigg[f(b) \cdot e^{-b/7} - f(3) \cdot e^{-3/7} \bigg] = -7\bigg[0 - 14 \cdot e^{-3/7}\bigg][/MATH]

 

[wassupman]

I was able to fiqure it out with your help. Thank you