I have a function with 6 variables (just for information I report it below, but I am satisfied with a theoretical answer): $$ a+ (a*b)/(1-b) + (a*c)/(1-c) + (a*d)/(1-d) + (a*e)/(1-e) + (a*f)/(1-f) + b*(c)/(1-b)*(a)/(1-b-c) + b*(d)/(1-b)*(a)/(1-b-d) + b*(e)/(1-b)*(a)/(1-b-e) + b*(f)/(1-b)*(a)/(1-b-f) + c*(b)/(1-c)*(a)/(1-c-b) + c*(d)/(1-c)*(a)/(1-c-d) + c*(e)/(1-c)*(a)/(1-c-e) + c*(f)/(1-c)*(a)/(1-c-f) + d*(b)/(1-d)*(a)/(1-d-b) + d*(c)/(1-d)*(a)/(1-d-c) + d*(e)/(1-d)*(a)/(1-d-e) + d*(f)/(1-d)*(a)/(1-d-f) + e*(b)/(1-e)*(a)/(1-e-b) + e*(c)/(1-e)*(a)/(1-e-c) + e*(d)/(1-e)*(a)/(1-e-d) + e*(f)/(1-e)*(a)/(1-e-f) + f*(b)/(1-f)*(a)/(1-f-b) + f*(c)/(1-f)*(a)/(1-f-c) + f*(d)/(1-f)*(a)/(1-f-d) + f*(e)/(1-f)*(a)/(1-f-e) $$ With the following constraints: $$ a+b+c+d+e+f=1 $$ $$ 0.07006139 <= a,b,c,d,e,f <= 0.3468013468 $$ The fact is that the function is increasing for all variables, and I have to show that to maximize it we must maximize $a$ ($a=0.3468013468$) and minimize the other variables ($b,c,d,e,f=(1-a)/5=0.13063972864$). This fact is a certainty, in fact having launched an optimization algorithm with Python I get these results, but I would need to be able to do a formal demonstration. Anyone know how I could set it?
Thanks in advance to those who will answer.