The fsolve worked.

Thank you!

Here's my variables:

E:=15.56*(Z+N)-17.23*(Z+N)^(2/3)-0.697*(Z^2)/((Z+N)^(1/3))-23.285*((Z+N)-2*Z)^2/(Z+N):
 Ep:=15.56*((Z-1)+N)-17.23*((Z-1)+N)^(2/3)-0.697*((Z-1)^2)/(((Z-1)+N)^(1/3))-23.285*(((Z-1)+N)-2*(Z-1))^2/((Z-1)+N):

I just tried the max method, and it didn't work. Here's what I did:

a:=plot(max(solve(Ep-E=0,N)),Z=0..20);
 Z:=5.852459350;
max(solve(Ep-E=0,N));

The plot list gave me the point ( among others ): [5.852459350, -1.649320198].

I used it in the Z value. the following line with the max gave me: 3.863675385

Why would the exact same command used in the plot give me -1.649320198, while when used outside the plot, it gives 3.863675385?