 The topic method of analysis and under this topic, we will study the method of Lagrange. Previously, we have studied that the optimization technique is mostly utilized in the economics and not even in the economics, in finance, in econometrics, in engineering to solve many solutions when we are having certain resources and when we have certain objective functions. So, when we say that a point that is called x-steric, here that we have taken and this x-steric is that point that we have attained from the point of optimization, means that is that point of value of x that satisfies our conditions of the constraints and it is able to provide us with the our objective function solution and this point will be the global minimizer if it fulfills the condition that this point is either less than or equal to then the function of x and that is the set of all other point or if we say in other words, all values of x in the entire interval which will be the lowest, it will be our global minimizer in the same way, but if I say that this point x-steric, if it fulfills our condition mathematically and if we look at it, it says that when our point t, functional value is less than or equal to the function of all other x values, but in a small neighborhood of x, so that will be local minimum and in the same way, if we say the next two points, then our global maximum will be there and if our local maximum will be there. So, if we look at these four conditions on the back end, then our mathematical rules and their conditions are present and if we look at it, then when we draw our curves, then because we have the data set in it, we have to take mathematical tools to solve it. Now, in the same mathematical tools, if I look at this point, if we are drawing a commodity over x axis and we are drawing a second commodity over y axis, and if I look at it mostly in microeconomics here, then we are taking it as that can be the consumption. So, if I say that on this side, we have a commodity where we are consuming x and on this side, if I say that we are consuming y, so if we look at it at the same point, then x and y combination, one value goes from here and it is less than or equal to here and this time x value is at this point, suppose it is 2, then we say that y value is at this point. When we have again x value is 4, then y value is this. So, like this, it shows us the various up and down movements. So, along this curve up and down movements, these points are that where we see that once the curve comes down and goes up again, if we look at these points, then mostly these points are called the inflection points. And if I look at it here, if we draw a production function mostly in microeconomics and we draw it in this form and if I draw a line from the origin from the center, then we see that it cuts this curve from here. So, if we look at it from here, then it means that this is the slope of the production function from here to here and if it goes up from here, then this point is called the inflection point. So, the meaning of keeping this example is that here y is the output and here capital or labour is our input. So, it means that this saddle point or inflection point can be present in any curve of our economics or in any graph of our mathematical term. Whenever a graph or a contour changes its slope, it changes, it gives a turning point. So, that point will be called the saddle point or if we look at it in another form, it will be called the inflection point and when we calculate its rate of change on that point, the rate of change in y with respect to x, so if we do it at this point, then if I say that change in y with respect to x at this point, then again change in y with respect to x. Now, what is the problem for our economists that we have general theoretical models when we have to prove various theories with the help of certain data but we also have incomplete data and it is possible that there are many variables like this that we do not have available data. So, if there is such a situation, then when there is no data available, then there is no numerical information or if there is data present, then it is in a qualitative form. In such a situation, it means that while solving the optimization technique, we will be careful because it will be able to give us a solution in that way but it may be difficult for us to calculate the saddle point and in the same way, if we look at it, then in our solution process, many of our general conditions will not be able to satisfy. What will be the shape of the objective function? We have a feasible set, whether it is convex or not, we have the shape of it properly because it is possible that the objective function will also come in the straight line and the feasible function will also come in the straight line and they both will be able to satisfy the objective function. So, the tendency will be available for both curves throughout, so how will we find out the best solution out of it? So, if we look at it in such a way, then we have some solution which is further in our economic subject. Math and statistics have been provided. We have two such saddle points or point of inflection. Through mathematics, we calculate its derivative or slope. The first condition is that the first derivative should be equal to zero. Either it is a maximum point or a minimum point. Its first derivative will be zero in every form. This means that the rate of change becomes zero and stops. That is the saddle point. In economics, the general theoretical models are there because we do not have any other solution to solve them if we do not have their data available. There may be a lot of data available in the model but it may be incomplete or we do not have the data in numerical form. In such a condition, it becomes very difficult for us to take mathematical or statistical tools for it and we say that we cannot solve it in the proper form. If we look at such a situation while solving it, then we have some conditions of optimization in which we say that those points, when we have to do maximum or minimum, then its two conditions are necessary. The first necessary condition is that the first derivative should be equal to zero. And the second derivative should be either positive or either negative. Now, if we talk about the positive and the negative, then if we draw a curve here, which we say that we have drawn in this form and here our point is y, this is our x and these two relationships are becoming something like this. So, we will see that we have this point of inflection or saddle point and on this point, change in y due to change in x, that should be equal to zero. So, the first derivative tells us that its slope is zero. But it doesn't tell us that either it is minimum or it is maximum. But when we take its second derivative and in the second derivative we say that now the rate of change should be positive. So, it means that now in the second derivative, change with respect to, if we say second, it should be not equal to zero. Rather, it should be greater than zero. If it is positive, then it will ascertain that this point is our minimum point. Similarly, if we draw another point like this, in which the relationship is like this, if we look at this point, then we will see that change in y with respect to x is zero. The first derivative will be zero. But the second derivative will show us that change in y with respect to x second, it is not equal to zero. Rather, it is less than means downward sloping. So, when after this point it is downward sloping, then we will decide that this is the maximum point. So, these two necessary conditions are very important for optimization. Similarly, we have a main assumption. We have a constraint problem that we have to see that while solving it, the constraint is really in the convex shape or not. So, to solve such situations, the next technique is called the Lagrange multiplier. And this Lagrange is an implicit function theorem or if we are in the second form, then it is based on an envelope theorem. And if we say in the short form, what is the envelope theorem? If we say that, when the light reaches directly without any media, then that path, the shortest path, is an implicit function that reaches the solution through the shortest path. And that is why we have an objective function. So, we find out its slope. On the other hand, if we look at the optimal point, then before reaching it, we find out the slope of the constraint. And when the point of tangency of these two slopes becomes the slope of the counter of the objective function, it should be equal to the slope of the budget or the constraint. So, we say that this will give us the tangency point. And if we look at this graphically, this is one function that is give us the constraint available with us. And if we say the constraint in the second form, then these are all available options. And this is our objective function that is present in front of us. And if we look here, then now we have a budget line. This is our tangency, the t line is basically touching this point of tangency. Because this is that point and if we look at this, then we have two lines that are present in front of us. This is our objective function and our constraint. The slope of these two slopes has become equal. So, the time when the slope is equal and the point touches it, it is called the point of tangency. And similarly, if we move this necessary condition forward in the form of leg range, it gives us that the first function minus lambda, if we take one equation, second and third. If we look here, then we have given a third equation here. So, in this, we see not only that we have choice variables, non-activity conditions, functional constraint, but with all these, a third equation that has become the condition is objective function. It is only function of x1 and x2. But this is our objective function in the form of leg range. So, this is x1, x2 and lambda. And now, we see this objective function. So, this is our objective function. And this is our constraint. And this is what we call in other form. So, this is our objective function. And we solve this and calculate the leg range multiplier. Now, if we want to calculate the leg range multiplier, how can we calculate it? So, if we look at any example, then we can solve it through it. If I want to solve this equation, then we will take first derivative, derivative of l with respect to x1. And then we will take derivative of l with respect to x2. Then we will take derivative of l with respect to lambda.