# Help maximizing this 14 variable scalar valued function?

I have never done multivariable calculus and the last time I did any calculus was 6 years ago in high school. This is just a problem related to something else I'm trying to solve just for fun.

I want to find the maximum where:

0 < a,b,c...m,n < 1 (the value of each variable is between 0 and 1)
and
a+b+c+....+m+n = 1

Here's the function:

                       (8644.23+2233.9055772*a)/(55.506*pi)*sin(55.506*pi*a/36)
+((276/142.65042+a)*2+b)*40.2462/pi*sin(55.506*pi*b/36)
+((276/142.65042+a+b)*2+c)*40.2462/pi*sin(55.506*pi*c/36)
+((276/142.65042+a+b+c)*2+d)*40.2462/pi*sin(55.506*pi*d/36)
+((276/142.65042+a+b+c+d)*2+e)*40.2462/pi*sin(55.506*pi*e/36)
+((276/142.65042+a+b+c+d+e)*2+f)*40.2462/pi*sin(55.506*pi*f/36)
+((276/142.65042+a+b+c+d+e+f)*2+g)*40.2462/pi*sin(55.506*pi*g/36)
+((276/142.65042+a+b+c+d+e+f+g)*2+h)*40.2462/pi*sin(55.506*pi*h/36)
+((276/142.65042+a+b+c+d+e+f+g+h)*2+i)*40.2462/pi*sin(55.506*pi*i/36)
+((276/142.65042+a+b+c+d+e+f+g+h+i)*2+j)*40.2462/pi*sin(55.506*pi*j/36)
+((276/142.65042+a+b+c+d+e+f+g+h+i+j)*2+k)*40.2462/pi*sin(55.506*pi*k/36)
+((276/142.65042+a+b+c+d+e+f+g+h+i+j+k)*2+l)*40.2462/pi*sin(55.506*pi*l/36)
+((276/142.65042+a+b+c+d+e+f+g+h+i+j+k+l)*2+m)*40.2462/pi*sin(55.506*pi*m/36)
+((276/142.65042+a+b+c+d+e+f+g+h+i+j+k+l+m)*2+n)*40.2462/pi*sin(55.506*pi*n/36)


I'm fairly certain that all the variables are simply 1/14 at the maximum, but I want to be sure. There might be an easier way to solve it but I can't really see it.

Wolfram alpha doesn't seem to be capable of taking a long function like this, and I couldn't find anything else...

Sorry if it's not simplified more, I already spent hours simplifying what it used to look like, and I didn't want to start rounding numbers when I have no idea how that will affect the output.

I found a slightly better solution. Your solution of all a,b,...=1/14 has an objective value of 2.961801e+002. Here is a solution I found using the global MINLP solver Couenne:

                           LOWER          LEVEL          UPPER

---- VAR a                   .             0.0765         1.0000
---- VAR b                   .             0.0756         1.0000
---- VAR c                   .             0.0747         1.0000
---- VAR d                   .             0.0738         1.0000
---- VAR e                   .             0.0730         1.0000
---- VAR f                   .             0.0722         1.0000
---- VAR g                   .             0.0715         1.0000
---- VAR h                   .             0.0708         1.0000
---- VAR i                   .             0.0701         1.0000
---- VAR j                   .             0.0695         1.0000
---- VAR k                   .             0.0689         1.0000
---- VAR l                   .             0.0683         1.0000
---- VAR m                   .             0.0678         1.0000
---- VAR n                   .             0.0672         1.0000
---- VAR z                 -INF          296.2082        +INF


The objective of this solution is 2.962082e+002.

I let the solver run overnight but it could not find a better solution. I was not able to prove there is no better solution, but I could prove that there does not exist a better solution than 3.026943e+002.