Optimization plays a major role in economic analysis. That's why this part of mathematics is absolutely important and we'll pay a lot of attention to it in this course. In order to understand optimization in the case of many variables, we have to recall what an optimization means for the functions of a single variable. So, let's start refreshing the definitions of a maximum and a minimum of such a function. So, suppose we have a function of one variable only and this function is defined on some interval ranging from a to b. So, if we consider a graph of the function, in order to refresh the concepts of maximum and minimum values. So, let's consider a graph like this. These are the end points but they're not included, a and b, because this is an open interval. At this point, which is X0, a function takes the value which is the greatest within the local neighborhood of this point, but at point X1 it takes the global maximum. So, let us formalize the concepts. So, we say that X0 is a maximum point of a function if there is some interval such that for all X from this interval, the values of the function are no greater than the value of the function taken at X0 point. So, this is a definition in the case of one variable of a local maximum. At X1, the function takes the greatest value if we compare the values from I interval. So then, in this definition we say that X1 is the global maximum because the value of the function taken at any point from the domain is no greater than the value taken at X1. So, this inequality holds for all values of X from the domain. Now, what tools do we use in order to find these points? By the way, talking about minimum, all we have to do, we need to reverse the signs over these inequalities. That's how we get the definition of a local minimum and respectively, a global minimum. So, the tools for finding maximum and minimum points using calculus. So, if we additionally know that the function f of x is continuously differentiable on I, then the necessary condition for a function to have maximum or minimum value at some points will look as follows, and it can be stated as a theorem. So, suppose a function f of x is continuously differentiable on I, it's written here. Let the point x* from I is either a maximum or a minimum of this function on I. Then, derivative at this point turn 0. This theorem provides all the necessary condition for a function to have a maximum or minimum value at a given point. We can provide an example of a function like y equals x cube whose derivative turn 0 at x equals 0. But this point is not maximum nor its minimum because we can draw the graph of a cubic parabola and clearly at 0 point is the point where the function increases. So, any solution of this equation f' equals 0 gives us x, which is a critical point of a function. In order to check whether a critical point is really a point of extremum, we need to perform a test, which is another theorem which provides a sufficiency condition for a maximum or a minimum. It is based on the analysis or the second order derivative of this function. So from now on, let's suppose that the function f of x is twice continuously differentiable on interval I. If we have a critical point and after the second differentiation we substitute this critical point into the second derivative, unless the value is 0, we can analyze the nature of this critical point based on the sign of the second order derivative. So, let's state it as a theorem. So, this is a theorem which provides sufficiency. So once again, the statement starts with the information regarding the differentiability of the function f of x. Let x* from this interval be a critical point of the function f of x. Then, given a negative value of the second order derivative at this point, that x* is a point of local maximum; given second order derivative, which takes a positive value, then x* is a point of local minimum.
