1.3.2  Problem Formulation Analytical solution is very difficult.  As in elementary calculus, where one attempts to extremize a function y=f(x) by solving  f '(x)=0, we try to "differentiate"  I(y), with respect to "y" in some sense to be explained below. Let y*(x) denote the optimum solution, i.e., y*(x) minimizes I(y).  Then any other function y(x) satisfies the constraints which may be represented by Above, (x) represents an arbitrary continuously differentiable perturbation function passing through (a,0) and (b,0).  Then we construct the "difference quotient" Consider now, the Taylor expansion of F(x,y,y') about the "point" (x,y*,y*' ) where Derivative of I at y* in the direction of is which is given by    Suppose we denote  by , because for fixed  is a function of  only.  Since , I(y) takes on its minimum value when . Hence, because f has a critical point (i.e., a minimum) at .