Question

How do you find the minimal cost of a cost function?

Answer 1

You (generally) use calculus. You find the first derivative of the cost function with respect to the variable you're interested in and solve. Substitute the answer back into the original cost function to find the minimal value (all this assumes the second derivative of the cost function at that value is positive (which implies a minimum function value is found there)).

Also, minima and maxima for functions may also be found at the boundary of the domain of the function. So if your cost function runs over all x in the set [0,1000], say, you should check the cost function at 0 and 1000 too.