Function dependent on Parameter defined by open Intervall

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SolidSnake

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Hi everyone,

I am new to the world of math forums and this is my first Question.
I ran into this problem I guess it is not to hard to handle but my math school days are long gone and I am having a hard time with this.

I got a fixed parameter say it is called A
There is a second Parameter B that is defined to be in the open interval (0.5A ; A)
And a third Parameter C defined to be in the open interval (0.5B ; B)

Now for my function.

f(x) = A*x + B*x + C*x

But it is also said, that B and C should be the smallest numbers Possible.

I don't know how to make calculations with this.

I can't substitute B for 0.5A as it is to not in the interval (0.5A ; A) but 0.50001A would be too large?

How could I do derivates and stuff like this?



Thank you very much!
 
Please clarify "But it is also said, that B and C should be the smallest numbers Possible. "

What's the problem with the derivative? This is a linear function.
 
If I understand correctly, he is asking for how to work with the open interval.
He cannot set a definite number, because open interval means that 0.5 is excluded. So, what is the smallest possible number under 0.5?
 
There is no "smallest number" in the open interval (x, y) nor in the half-open interval (x, y]. Of course, in the closed interval [x, y] or the half-open interval [x, y) the smallest number is x.
 
I think the big question is, where do the requirements come from? What is the context of the question?

We know that there is no smallest number satisfying the conditions; rather, the result is simply known to be greater than A*x + (0.5A)*x + (0.5(0.5A))*x = 1.75x.

So, why do you need the smallest? Why must the intervals be open? And how does this relate to derivatives?
 
Sorry I think I was not precise enough.

A is any real number.

B must be greater than A*[1/(1+x)]

C must be greater then B*[1/(1+x)]

My function is

A*x+B*x+C*x

I want to calculate the derivative of this function in order to analyse its slope at any point

I just can't wrap my mind around these two Intervalls.

In the limit case were B is equal to A*[1/(1+x)]

and

C equal to B*[1/(1+x)]

I can substitute C = B*[1/(1+x)] = A*[1/(1+x)]*[1/(1+x)] = A*[1/(1+x)]^2

Than It is easy to do all calculations derivatives etc.

But how do i handle these open intervals?
 
"Not precise enough"? You didn't even mention what you really wanted to do!

But it's still very unclear. Are A, B, and C supposed to be constants? Then they can't depend on x. Or is the x in [1/(1+x)] different from the other x, with respect to which you are going to differentiate?

I think we need further background. What is the source of the restrictions on B and C? And why do you need a minimum?

I suspect the problem needs to be described differently, especially if it's true that B and C are themselves functions of x. Maybe, for example, you need to replace B with [A/(1+x)]*(1+epsilon) or something, and take a limit as epsilon approaches zero.
 
A is a constant and B and C depend on A which is a constant and on x which is the same x as the from the function i want to differentiate.

"replace B with [A/(1+x)]*(1+epsilon)"

already calculated the derivative and the root of it for epsilon = 0

But i don't know how to handle the epsilon
 
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