# Factoring a number with Euler's constant involved?

## Homework Statement

Determine the maximum value of the function f(x) = 4e^(-x) - e^(x)

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## The Attempt at a Solution

1. I first found the derivative of the function and it turned out to be f`(x) = -4e^-x - e^x
2. I believe that to find the maximum value of this function, you need to find out values of x that would equal to 0. (Critical numbers)
Then after finding these crit numbers, you can judge by looking at them and see which value returns to be the highest.
My problem is i do not know how to factor this expression with euler's constant involved. Can anyone help? (I think you can factor e out, but then i dont get a smooth expression or any critical numbers.)

Edit: I just realized i wrote in the title to be number, but i meant expression.

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Edit: iam not sure if this is the way to determine global extrema, so if otherwise someone please explain

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LCKurtz
Homework Helper
Gold Member
Factor an $e$ out??? Were you given any particular interval for $x$? Looks to me like you are going to have trouble getting that derivative to equal zero.

Yes that is what im trying to factor.
Say the interval was (-1 < x < 2)

How would you go about the problem given the interval?

Edit : Sorry I meant to say that I tried factoring the e out but since the exponents wasn't the same i wasn't sure if i was correct. Can you factor -e^-x out?

LCKurtz
Homework Helper
Gold Member
Yes that is what im trying to factor.
Say the interval was (-1 < x < 2)

How would you go about the problem given the interval?

Edit : Sorry I meant to say that I tried factoring the e out but since the exponents wasn't the same i wasn't sure if i was correct. Can you factor -e^-x out?
You should write out what expression you are talking about. I guess it is the $f'(x)=-4e^{-x}-e^x$. If you remember that $e^{-x}=\frac 1 {e^x}$, what do you think about your factoring? Here are some other things to think about:
1. Do you think $f'(x)=0$ is possible?
2. What theorems do you know about extrema on finite intervals?
3. Does it matter if the interval is $-1<x<2$ versus $-1\le x \le 2$?

just sub in -1 in the original function then?

i don't need to do anything else i believe

LCKurtz