Let f of x be a continuous function defined on some close segment from a to b. Then, there exist m lowercase, and a capital letter M, real numbers, such that the function is bounded from above and from below for all x from a to b. More over, there exist x one, x two points from this segment, such that. So, both values, the maximum value of the function, and the minimum value of the function, they're both obtained on this close segment. But, we need an analog of such a theorem for the n-dimensional case to begin with. The close segment should be changed into a compact set. So, let me remind you, the definition of a compact set, or simply compact. This is a reminder of the definition, a set S, from n-dimensional space is called compact, or compact set, if it is bounded and closed. By the way, and the problem we've been dealing with, and we stopped at the point when we have to classify the found critical points. The constraints set, this is a unit circle into the plane, and clearly it's bounded, and also it's closed because whenever we choose a convergent sequence of points belonging to the unit circle, its limit belong to the same circle. That means that the function in question, we call it objective function, which was z equals three x plus four y, which is clearly continuous at any point, takes the greatest and least values on the constraints set. So, we can refer to the analog or the theorem of Weierstrass, which actually this statement for the n-dimensional case looks very similar with the only exception that we change the segment from a to b to compact set. The rest is the same. So, the function which is continuous on the compact set is bounded from above, bound from the below, and there exists two points from the set, at which these values are retained. That means that the critical points we have found, we have found two points are exactly the points of the maximum and minimum. So, that means that A is maximum and B is minimum. Unfortunately, we don't have the luxury or working with a compact sets all the time. For example, let's consider a problem of the costs minimization in the form of, so, this example is drawn from microeconomics production theory. We are minimizing costs. These are the costs of labor, costs of capital, and that should be minimized subject to the production function constraint. So, q is the output of the function and this is a Cobb-Douglas function, k times l, where k and l takes non-negative values. So, in order to have at least a graphical approach for solving the problem, let's consider the space of vectors that will be the labor axis and this is the capital X's. So, the graphical representation of the constraint is called an isoquant and it looks like a hyperbola. So, we are looking for optimal values of labor and capital for which the costs takes the least value. If we fix the value of the costs, we get a straight line, which may cross the isoquant, or may touch it. If it touches at some point somewhere here, that will be the solution of the problem, graphical solution of the problem, but strictly speaking since the isoquant is not bounded, it's not a compact set. So, we cannot employ the Weierstrass theorem in this case. We need to find some other tool which will help us to classify the critical points or the Lagrangian, and we'll do it right now.