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girl_in_jeans
S.O.S. Newbie
Joined: 12 Nov 2003
Posts: 3
Location: Netherlands
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 Posted: Tue, 25 Nov 2003 16:26:57 UTC Post subject: Smallest and largest values |
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I have the function f(x) = x^4 * e^(-x)
I have found the maxima to be x=4 and the minima to be x=0. As well as finding these points the question asks for smallest and largest values. The answer is that the smallest value is 0 and the largest doesn't exist. Why doesn't it exist and how do they get that? |
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Soroban
Member of the 'S.O.S. Math' Hall of Fame
Joined: 19 May 2003
Posts: 7246
Location: Lexington, MA
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| Posted: Tue, 25 Nov 2003 18:56:48 UTC Post subject: Re: Smallest and largest values |
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Hello, girl_in_jeans!
| Quote: | I have the function 
I have found the maxima to be x = 4 and the minima to be x = 0. |
These are correct!
| Quote: | As well as finding these points the question asks for smallest and largest values.
The answer is that the smallest value is 0 and the largest doesn't exist.
Why doesn't it exist and how do they get that? |
The function is  . . . which is always positive.
It can be shown that: 
That is, as x gets larger, the graph approaches the x-axis asymptotically.
Therefore, the minimum you found (0,0) is the absolute minimum.
But 
Baby talk: As x gets "more negative",  gets larger.
Therefore, the graph rises to the left, without bound . . . there is no absolute maximum. |
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girl_in_jeans
S.O.S. Newbie
Joined: 12 Nov 2003
Posts: 3
Location: Netherlands
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| Posted: Wed, 26 Nov 2003 15:49:13 UTC Post subject: |
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Thank you Soroban for your your explanation! It's very helpful. I just have a further question.
You said that when the limit (of my original function) is infinity (+) it is 0. When I can't work this out with a graph-drawing calculator, can I say that it is 0 because there is x in the denominator? Am I thinking correctly?
What about when it is infinity(-)? Do I need to find the extreme points first and go from there? |
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Soroban
Member of the 'S.O.S. Math' Hall of Fame
Joined: 19 May 2003
Posts: 7246
Location: Lexington, MA
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| Posted: Wed, 26 Nov 2003 18:05:46 UTC Post subject: |
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Hello, girl_in_jeans!
| Quote: |
You said that when the limit (of my original function) is infinity (+) it is 0.
When I can't work this out with a graph-drawing calculator,
can I say that it is 0 because there is x in the denominator?
Am I thinking correctly? |
There's more to consider than just "x in the denominator".
With  , there is an x in the denominator
which makes the denominator get larger and larger.
But there's also an x in the numerator.
So the numerator gets larger and larger, too.
So what happens to the fraction?
It can be shown that  "grows faster" than 
so that the fraction gets smaller and smaller, and approaches 0.
So, if there are x's in the top and bottom, what happens to the fraction
depends on which function is more "powerful".
| Quote: | What about when it is infinity(-)?
Do I need to find the extreme points first and go from there? |
We should always locate the extreme points.
But we don't need them to consider negative infinity.
We have: 
Suppose x is a negative number.
Let  . . . where a is a positive number.
Then the function is: 
And we can see that this function grows without bound. |
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