These two theorems are quite useful. I'll provide an example how one of these theorems can be applied. Let's consider a problem when we are trying to figure out whether a particular set is a convex set. This is our set A in a coordinate plane, and we can draw by shading. That's this that's a hyperbola given by the equation y = 1/ x. The shading will provide this set A. Okay. Check that A is convex. We easily see that A is epigraph of the function f where f(x) = 1/x defined on the positive half x-axis. So, x is greater than zero. So if the function f(x) is a convex function, then, according to the theorem, the epigraph of this function f should be a convex set. How to check whether it's convex: in one-dimensional case, it's done very easily by finding the second order derivative of this function. So we have a positive derivative because we get 2/x^2, it's strictly positive, and applying this condition, we draw a conclusion that the function is convex and that means the epigraph of this function is convex and that entails the fact that A is a convex set. Now, I would like to show one of the inequalities for a concave function which is very useful for microeconomic applications. Let's start firstly with the one-dimensional case. I will draw a typical graph of the concave function, so y = f(x) is concave and morrow function belongs to C^1 clause. So we can differentiate it and its derivative is continuous everywhere, and that's the graph. So if I choose some point here on the x-axis and label it with x*, I would like to draw the tangent line to the graph or this function passing through the point and touching the graph of this function at this point. Now, if I choose some other point, I'm choosing to the right of x*, but it's not necessary. I can choose absolutely with the same reason to the left, but this is my choice of x to the right. Then, I would like to compare the values of the function that the function takes at x with the value of the linear function which represents the graph of the tangent line. So we need to compare if I also draw the horizontal line passing through the point that will be y*. So I'm comparing the segments and it's clear that the rise in the value of the function because I'm saying rise because the function is monotonic, it's increasing. So this rise in the value of the function is no greater than the product of the derivatives by the difference in x-values. That is clear from the drawing clearly because I'm comparing this is delta f with the df first order differential of the function f. But clearly, well, you can easily find in texts the rigorous proof of this formula. It can be rewritten in a different form. F(x) is no greater than L(x, x*), where we by capital L define the linear function which provides the equation of the tangent line. And, by this inequality, we emphasize the fact that given a concave function, the graph of the concave function lies below the graph of the tangent line which touches the graph of f. This inequality can be generalized for the case of n-dimensional space, but this time instead of an ordinary derivative, we'll be dealing with the gradient of the function f taken at x*. And instead of multiplying two numbers, we will be fine using the inner product of two vectors. One of them being the gradient, the second is the increment in independent variable being a vector. And, that can be formulated like a theorem. So let's consider a theorem. Also, it will be provided with no proof. Although I will say that the proof is based on the inequality of one-dimensional case and it's not quite difficult to prove it. So, let f(x), a continuous differentiable function be defined on S where S is an [inaudible] set, open and also convex set in R^n. F(x) is concave in S if and only if for each x, x* from S the following inequality holds, and that's our inequality. The sign of an inner product, here we put the gradient, we call it nabla f(x*)(x - x*).