 Hello and welcome to the session. In this session, we will discuss about critical points and extreme points of a function. First of all, let us discuss critical points. Now, in our earlier session, we have discussed about increasing and decreasing functions. Now, the points in the domain of a function from where the graph of the function changes from increasing to decreasing or from decreasing to increasing are called critical points. Now, here you can see graph of the function f of x is equal to x square. Here you can see that this function is decreasing for x is less than 0 and increasing for x is greater than 0. So, for x is equal to 0, the graph changes from decreasing to increasing. So, x is equal to 0 is the critical point of this function that is the function f of x is equal to x square. Now, let us draw a horizontal line at x is equal to 0. So, we have drawn this horizontal line at the point x is equal to 0 and we see that this line is tangent to the curve at this point. So, critical points are those points on a graph at which a line drawn is tangent to the curve. Now, a polynomial may possess three types of critical points. First is maximum, second is minimum and third is point of inflection. Now, one critical point is maximum or minimum. Now, suppose we have a critical point at x is equal to c and we want to check that whether it is a point of maximum or minimum. Let us check it graphically. Now, if the function is increasing to the left of x is equal to c and decreasing to the right of x is equal to c, then the function is maximum at x is equal to c. Now, we are going to see the graph of the function f of x is equal to minus x square plus 4x plus 1. Now, here we can see when we draw a horizontal line at the point x is equal to 2, then it will be tangent to the curve at x is equal to 2. So, x is equal to 2 is the critical point. Also, the curve increases when x is less than 2 and it decreases when x is greater than 2. So, the curve changes from increasing to decreasing. It means at x is equal to 2, the function f of x is maximum. Now, if the function is decreasing to the left of x is equal to c and increasing to the right of c, then the function is minimum at x is equal to c. Now, here we can see graph of the function f of x is equal to x square. Now, here clearly you can see that x is equal to 0 is the critical point and also you can see that the curve decreases when x is less than 0 and it increases when x is greater than 0. So, the curve changes from decreasing to increasing. At x is equal to 0, the function f of x is minimum. Now, let us find maximum and minimum algebraically. Now, if x is equal to c is the critical point, we consider the points c minus h and c plus h where f is very small taken in decimals. Now, c minus h will lie to the left of c and c plus h will lie to the right of c to limit the value of the function f of x at the point x is equal to c minus h and at the point x is equal to c plus h. Now, if f of c minus h is less than f of c and f of c plus h is less than f of c when the function is maximum at x is equal to c. Now, in this graph you can see that f of c minus h is less than f of c and f of c plus h is again less than f of c. So, now the function is maximum at x is equal to c. Now, f of c minus h is greater than f of c. f of c plus h is also greater than f of c when the function is minimum at the point x is equal to c. For example, for the function f of x is equal to minus x square plus 4x plus 1 we want to determine whether the critical point x is equal to 2 is maximum or minimum. You will consider a point very close to the point x is equal to 2 but lying to the left of 2 let it be 1.9. Now, this is the point c minus h where c is equal to 2 and x is equal to 0.1. Now, we will consider a point very close to x is equal to 2 but lying to the right of 2 let it be 2.1. Now, this is the point c plus h where h is again 0.1. Now, using calculator f of 1.9 is equal to 3.99 and f of 2.1 is equal to 3.99. So, we have obtained the values of f of c minus h and f of c plus h. Now, f of c is equal to f of 2 and on calculating this is equal to 5. Now, here we see that f of c minus h that is f of 1.9 is less than f of c that is f of 2 and f of c plus h that is f of 2.1 is again less than f of c that is f of 2. So, at x is equal to 2 f of x is maximum. Now, if at the point x is equal to c f of x is maximum or minimum then the point of maxima or minima is given by the ordered pair c f of c. So, for this example maximum point is given by the ordered pair 2 f of 2 that is 2 that is the ordered pair 2 and f of 2 is 5. So, maximum point is given by the ordered pair 2 5. This is also called the extreme point. Now, let us discuss what is extrema. Now, these are the values of the function at each maximum and minimum point. Now, let us see what is point of inflection. Now, point of inflection is the point where graph changes its curvature. Now, in this graph at this point the graph changes its curvature. It is neither maximum nor minimum point. As regularly, if f of c minus h is less than f of c and f of c plus h is greater than f of c or if f of c minus h is greater than f of c and f of c plus h is less than f of c then x is equal to c is point of inflection. Now, a function can have more than one critical point. In this case, we find the maximum and minimum points at each critical value. So, in different intervals the function will be maximum or minimum and if there are more than one minimum or maximum points in different intervals of a function then we find the absolute maximum and absolute minimum value of the function. Now, of all the maximum points, the greatest value that the function assumes over its domain is called the absolute maximum, similarly, of all the minimum points the least value the function assumes over its domain is called the absolute minimum. Now, consider the following graph of a function. Here we have three critical points that is at x is equal to 2, at x is equal to 4 and at x is equal to 6 and we have two minimum points that is at x is equal to 2 and x is equal to 6. It means in some interval x is equal to 2 is minimum and in some interval x is equal to 6 is minimum and these are called relative minimum Thus, relative minimum is least value of function on some interval of the domain Similarly, relative maximum is maximum value of function on some interval of the domain. So, in this session we have discussed about critical points and extreme points of a function and this completes our session. Hope you all have enjoyed the session.