Maximum of exponential function

AI Thread Summary
The discussion focuses on finding the maximum of the function m=n*e^(-nt) and demonstrating that it occurs at m=1/(t*e). Participants explore methods to derive this maximum analytically rather than graphically, emphasizing the importance of the first derivative and the product rule in differentiation. There is confusion regarding the appearance of e in the denominator and the implications of setting e^(-nt) to zero. Ultimately, the correct substitution of n=1/t into the original equation is identified as the key to resolving the issue. The conversation concludes with a clarification that resolves the misunderstanding about the calculations.
Binder12345
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Homework Statement


given the formula m=n*e^(-nt) show that the maximum of this curve is at m=1/(t*e^(1)).

2. The attempt at a solution
I can show this graphically but I am curious if it is possible to do it by hand?
 
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So the "t" here is a fixed but unknown constant and n is the parameter we are adjusting, right? We have a function of one variable and are looking for its maximum value.

If a function has has a maximum, what can we say about its first derivative at that maximum?

If we want to prove that it has a maximum without graphing it, are there any theorems that we might be able to invoke?
 
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If we set it equal to zero and solve that will be our maximum
 
If you write it as ## y=xe^{-tx} ##, and use the hint by @jbriggs444 , it should be straightforward.
 
That all makes sense my issue is how do I get the e^1 in the denominator? because isn't e^(-nt)= 0 a non real answer?
 
Binder12345 said:
That all makes sense my issue is how do I get the e^1 in the denominator? because isn't e^(-nt)= 0 a non real answer?
You must be differentiating the expression wrt n incorrectly. You need the product rule. If still stuck, please post all your working.
 
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This is strange... transiently there was another post by @Binder12345 (I think) with the right answer, then it disappeared.
 
haruspex said:
You must be differentiating the expression wrt n incorrectly. You need the product rule. If still stuck, please post all your working.
Sorry was going to edit and accidentally deleted :


I get:
(1-nt)e^(nt)

set equal to 0 and solve:
(1-nt)e^(nt)=0 -> 1-nt=0 -> n=1/t

I'm missing my e^1 in the denominator though
 
Binder12345 said:
missing my e^1 in the denominator though
Then you are going wrong substituting n=1/t into the original equation.
 
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haruspex said:
Then you are going wrong substituting n=1/t into the original equation.
Yup that is exactly what I was doing wrong! :\

Thank you
 
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