It is NEVER a good idea to start any optimization at zero. That may easily screw up the gradient estimation, in case the code uses a relative increment on the variables. A random starting point, when you don't have a good idea of a guess, is always safer. Best of course is to use a start point that is close to the result.
No need to use an m-file function here. A function handle is sufficient.
F = @(x) [(39*x(1)^2*x(2)^2)/100 - (x(2)^2)/5 - 2*x(1)^2 + 1;(39*x(1)^2*x(2)^2)/100 - (7*x(2)^2)/25 - (14*x(1)^2)/5 + 49/25];
[x,fval] = fsolve(F,rand(1,2))
fsolve completed because the vector of function values is near zero
as measured by the default value of the function tolerance, and
the problem appears regular as measured by the gradient.
<stopping criteria details>