Question Regarding Limits approaching Infnity

[LeviaThan]

Hello guys, I have a problem regarding this particular question which states:

Consider the function f(x) = xe^-bx, where b > 0 is a constant and x is all real numbers.

a) Use limits to describe the long-term behavior of this function?
b) find all local maximum and minimum values of f(x)?


Thank you guys, I really do appreciate your assistance...

 

[stapel]

Consider the function f(x) = xe^-bx, where b > 0 is a constant and x is all real numbers.

a) Use limits to describe the long-term behavior of this function?

What is meant by "long-term behavior"? Is your book implying that you're supposed to find the limits as x gets very large (in each of the positive and negative directions)? If not, then what is your book asking? If so, then what are your thoughts? What have you tried? How far have you gotten? Where are you stuck?

b) find all local maximum and minimum values of f(x)?

What did you get when you followed the usual procedure of taking the derivative and setting it equal to zero, etc?

Please be complete. Thank you!