[updated 6.12.2023]
For optimization problem, the optimum solution of (x1,x2) is found by setting the derivative of objectives of f(x1,x2) wrt x1 and x2 = 0, i.e., df(x1,x2)/dx1 = 0 and df(x1,x2)/dx2 = 0 We can use Newton method to solve such system equation of 2 variables as above. The Jacobian matrix has its specific name as the Hessian matrix in the optimization problem, but they have the same meaning.