 Hi and welcome to the session. Today we will learn about roles and mean value theorem. So first of all let us start with roles theorem. Let the function f from the closed interval a comma b to the set of real numbers v continuous on closed interval a comma b and differentiable on open interval a comma b such that f of a is equal to f of b where a and b are some real numbers. Then there exists some c in open interval a comma b such that f dash of c is equal to 0. Now let us see the geometrical interpretation of roles theorem. Here suppose we are given the graph of the function y equal to f of x. Suppose that roles theorem is applicable on the given function. Now as we know that the slope of the tangent at any point on the graph y equal to f of x is the derivative of f of x at that point. Now according to roles theorem there exists a point c which belongs to open interval a comma b that is in between the numbers a and b such that f dash of c is equal to 0. That means the slope of the tangent to the curve at point c is 0. So here we have the point c in between a and b such that the slope of the tangent to the curve y equal to f of x at point c is 0 that is here f dash of c is equal to 0. That means the tangent at point c is parallel to x axis. Thus this is the claim of the roles theorem that in the graph the slope of the tangent becomes 0 at least at one point. Let's take one example. Here we are given the function f of x equal to x square plus 3x minus 4 and we need to verify the roles theorem for x belonging to the closed interval minus 4 comma 1. In this as we can see that f of x is a polynomial function. So this implies that f of x is continuous on the closed interval minus 4 comma 1. Also f dash of x is equal to 2x plus 3. So from here it is clear that f of x is differentiable on the open interval minus 4 comma 1. Now let us find f of minus 4 and f of 1. So f of minus 4 is equal to 0 and f of 1 is also equal to 0. So this implies f of minus 4 is equal to f of 1. So now we can say that all the conditions of roles theorem are unsatisfied. Thus by roles theorem there exists some c belonging to the open interval minus 4 comma 1 such that f of c is equal to 0. So now let us substitute f dash of c equal to 0 and from this we will get 2c plus 3 is equal to 0 and from this we will get the value of c as minus 3 by 2 which belongs to the open interval minus 4 comma 1. So that means roles theorem is verified. Now let us move on to mean value theorem. Let the function f from the closed interval a comma b to the set of real numbers be a continuous function on the closed interval a comma b and differentiable on open interval a comma b then there exist in the open interval a comma b such that dash of c is equal to f of b minus f of a upon b minus a. Now here is the geometrical interpretation of mean value theorem. Suppose we are given the graph of the function y equal to f of x and also suppose that mean value theorem is applicable to the given function. So according to mean value theorem there exist some point c in the open interval a comma b that is in between the points a and b such that f dash of c is equal to f of b minus f of a upon b minus a. Now suppose this is the point a and this is the point b. So this is the secant a b and as we can notice that f of b minus f of a upon b minus a is the slope of the secant a b and f dash of c is the slope of the tangent to the curve at point c comma f of c. That means the slope of the tangent to the curve at point c comma f of c is equal to the slope of the secant a b or we can say that the secant a b is parallel to the tangent to the curve at point c comma f of c that means there will be a point c in between a and b such that the tangent at that point will be parallel to the secant a b. So this is the point c and the tangent at point c comma f of c is parallel to the secant a b. So I hope mean value theorem must be clear to you. Now let's take one example for this. Here we need to verify mean value theorem for the function f of x equal to x into 2 minus x where x belongs to the closed interval 0 comma 1. We can rewrite f of x as 2x minus x square. Now as we can see that f of x is a polynomial function. So this implies f of x is continuous on the closed interval 0 comma 1. Also f dash of x is equal to 2 minus 2x. So from here it is clear that f of x is differentiable on the open interval 0 comma 1. So with this all conditions of mean value theorem satisfied thus by mean value theorem there exists in the open interval 0 comma 1 such that f dash of c is equal to f of 1 minus f of 0 upon 1 minus 0 and from this we get 2 minus 2c is equal to 1 minus 0 upon 1 and this gives us c equals to 1 by 2 which belongs to the open interval 0 comma 1. Thus mean value theorem is verified. So in this session we have learned role theorem and mean value theorem. With this we finish this session. Hope you must have understood all the concepts. Goodbye, take care and have a nice day.