 Hello and welcome to the session. In this session we will discuss a question which says that which of the given functions have larger maximum? f of x is equal to minus 3 into x minus 3 whole square plus 2 or f of t is equal to minus 8 t square plus 16 t. Now let us start with the solution of the given question. Here we are given these two quadratic functions that is f of x is equal to minus 3 into x minus 3 whole square plus 2 and f of t is equal to minus 8 t square plus 16 t. And we have to find that which of these functions has larger maximum? Let us see the first function that is f of x is equal to minus 3 into x minus 3 whole square plus 2. Now we know that vertex form of quadratic function is given by a into x minus h whole square plus k where vertex that is coordinates of vertex are h k. Now comparing this given function with vertex form of a quadratic function we have h is equal to 3 and k is equal to 2. So vertex of this function is given by the ordered pair h k that is the ordered pair 3 2 also here you can see that a is equal to minus 3. Now as a is equal to minus 3 which is less than 0 so graph of a given function will open downwards and vertex will be the maximum point and while coordinate of the vertex will give the maximum value of the function. Now here you can see while coordinate of the vertex is 2 so the maximum value of the function is equal to 2. Now let us take the second function that is h of t is equal to minus 8 t square plus 16 t. Now first of all we will express this quadratic function in vertex form by method of completing the squares. So this implies h of t is equal to now here first of all we will make coefficient of t square 1 and this we will take minus 8 common from where these terms and it will be minus 8 into t square minus 2 t the whole. So this implies h of t is equal to minus 8 into now where within the vertex we will add and subtract square of half the coefficient of t so it will be t square minus 2 t now half the coefficient of t is minus 1 so its square is 1 so here we will add 1 and we will subtract 1. Further this implies h of t is equal to minus 8 into now here t square minus 2 t plus 1 is t minus 1 whole square minus 1 and this complete well. This further implies h of t is equal to minus 8 into t minus 1 whole square plus 8. So we have expressed this function in vertex form on comparing here we have h is equal to 1 and k is equal to 8. So vertex of this quadratic function is given by the ordered pair 1, 8 also here you can see that a is equal to minus 8 which is less than 0. So graph of this function will open downwards and vertex will be the maximum point and while coordinate of the vertex will give the maximum value of the function now here while coordinate of vertex is equal to h so maximum value of the function is equal to 8. Now we have obtained that the maximum value of the function f of x is equal to 2 and the maximum value of the function h of t is equal to 8 thus the function h of t has larger maximum now let us see this graphically here this red curve is the graph of the function f of x and this blue curve is the graph of the function h of t now here this point is the highest point of this red curve and this point is the highest point of this blue curve clearly you can see that the function h of t has larger maximum than the function f of x so this is the solution of the given question that's all for this session hope you all have enjoyed this session