 Hi and welcome to the session. Let us discuss the following question. Question says find the absolute maximum value and absolute minimum value of the following functions in the given intervals. Function F is given by fx equal to 4x minus 1 upon 2 x square where x belongs to close interval minus 2 comma 9 upon 2. First of all let us understand that for finding absolute maximum or absolute minimum values of a function in a given closed interval. First of all we will find all critical points of function F in the given interval. Then we will find the value of F at all critical points and at the end points of the interval. Now we will identify the maximum and minimum values of function F out of these values which we have calculated in step 2. The maximum value will be the absolute maximum value of function F and the minimum value will be the absolute minimum value of function F. This is the key idea to solve the given question. Let us start the solution. Function F is given by fx equal to 4x minus 1 upon 2 x square where x belongs to closed interval minus 2 9 upon 2. Now we will find out f dash x f dash x is equal to 4 minus 1 upon 2 multiplied by 2x. There we have differentiated both sides with respect to x below derivative of 4x is equal to 4 and derivative of 1 upon 2 x square is equal to 1 upon 2 multiplied by 2x. Now 2 and 2 will cancel each other and we get f dash x equal to 4 minus x. Now to find critical points we put f dash x equal to 0. This implies 4 minus x is equal to 0. Now this further implies 4 is equal to x. Adding x on both sides we get 4 is equal to x or we can simply write it as x equal to 4. Now clearly we can see 4 lies in this interval. So we will find the value of f at x equal to 4 at x equal to minus 2 and at x equal to 9 upon 2. So first of all let us find out f minus 2. It is equal to 4 multiplied by minus 2 minus 1 upon 2 multiplied by minus 2 square. Now simplifying we get minus 8 minus 1 upon 2 multiplied by 4 we know minus 2 square is equal to positive 4. Now we will cancel common pattern 2. Now we get h minus h minus 2 this is equal to minus 10. So we get value of f minus 2 s minus 10. Now we will find the value of the function at x equal to 4. This is equal to 4 multiplied by 4 minus 1 upon 2 multiplied by 4 square. This is equal to 16 minus 1 upon 2 multiplied by 16. We know 4 multiplied by 4 is equal to 16. 4 square is equal to 16. Now we will cancel the common factor 2 and get 16 minus 8. Now f 4 is equal to 8. Now we will find the value of the function at x equal to 9 upon 2. f 9 upon 2 is equal to 4 multiplied by 9 upon 2 minus 1 upon 2 multiplied by 9 upon 2 square. Now on simplifying we get f 9 upon 2 equal to 18 minus 1 upon 2 multiplied by 81 upon 4. We know 9 upon 2 square is equal to 81 upon 4. Now this is further equal to 18 minus 81 upon 8. Now subtracting by taking LCM we get 144 minus 81 upon 8. Now this is equal to 63 upon 8. So we get f 9 upon 2 is equal to h minus 1 upon 2 is equal to 63 upon 8 which can be further written as 7.875. Now clearly we can see minimum value of function f is equal to minus 10. So absolute minimum value of function f is equal to minus 10 which occurs at x equal to minus 2 and maximum value of function f is equal to 8 which occurs at x equal to 4. So we can write absolute minimum value of function f on interval minus 2 comma 9 upon 2 is minus 10 occurring at x equal to minus 2 and absolute maximum value of function f on interval minus 2 comma 9 upon 2 is 8 occurring at x equal to 4. So this is our required answer. This completes the session. Hope you understood the solution. Take care and keep smiling.