Hello,

I've been stuck with a math problem for a while, so I decided to ask for help. I have a function :

Y = [tex]\frac{mx}{k + x}[/tex]

25 [tex]\le[/tex] m [tex]\le[/tex] 45

0 [tex]\le[/tex] k [tex]\le[/tex] 4000

0 [tex]\le[/tex] Y [tex]\le[/tex] 25

0 [tex]\le[/tex] x [tex]\le[/tex] 3000

x and Y are integers

I can calculate Y for any x by doing an experiment. The values of m and k are determined by 6 numbers [tex]z_{1 }[/tex],[tex]z_{2 }[/tex],[tex]z_{3 }[/tex],[tex]z_{a }[/tex],[tex]z_{5 }[/tex],[tex]z_{6 }[/tex]. I can set the values of them before the experiment.The sum of these numbers must be equal to 9000 :

[tex]\sum_{a=1}^{6 } z_{a }[/tex] = 9000

All z are integers. They are all interchangeable, for example k (1000,1000,1000,1000,1000,4000) = k (1000,1000,1000,1000,4000,1000) = k (1000,1000,1000,4000,1000,1000) and so on.

The value of coefficient k depends on the distribution of numbers z, for example : k (1500,1500,1500,1500,1500,1500) [tex]\ne[/tex] k (1,1,1,1,1,8995). I want to know the law how k depends on the distribution of numbers z.

I've found out that k reaches the highest value of roughly 4000 when z are distributed at equal values of 1500. As I change numbers z, k decreases, and k reaches zero if 4 of 6 numbers z are set to zero.

( then the function Y = 25 ) The coefficient m behaves in the same manner, but differs from k in limits.

The first thing I tried to do is to increase one z slightly from the point k (1500,1500,1500,1500,1500,1500) by [tex]\triangle z_{1 }[/tex] and find the dependency [tex]\triangle[/tex]k ([tex]\triangle z_{1 }[/tex]), but if I increase [tex]z_{1 }[/tex] by [tex]\triangle[/tex]z I must decrease other z by [tex]\triangle z_{1 }[/tex] because [tex]\sum_{a=1}^{6 } z_{n }[/tex] = 9000.( I can also decrease 2 numbers z by [tex]\frac{1}{2} \triangle z_{1 }[/tex] or 1 number z by [tex]\frac{1}{3} \triangle z_{1 }[/tex] and the other z by [tex]\frac{2}{3} \triangle z_{1 }[/tex] and so on.)

I found that function [tex]\triangle[/tex]k = [tex]\triangle[/tex]k( 1500 + [tex]\triangle[/tex]z,1500 - [tex]\triangle[/tex]z,1500,1500,1500,1500) approximates to polynom :

[tex]\triangle[/tex]k = a[tex]\triangle z^{3}[/tex] + b[tex]\triangle z^{2}[/tex]+cz + d

I do not know what to do next. My goal is to find function k = f (z). Any advice?