 Hi and welcome to the session. I am Shashi and I am going to help you with the following question. Question says, find the maximum value of 2x cube – 24x plus 107 in the interval 13. Find the maximum value of the same function in closed interval – 3 – 1. First of all, let us understand for finding absolute maximum or absolute minimum values of function f in closed interval a, b. First of all, we will find all critical points of function f in the interval a, b. Now, we will find value of function f at all critical points and at the end points of the given interval. Now, identify the maximum and minimum values of function f out of the values which we have calculated in step 2. The maximum value of function f is absolute maximum value of function f and minimum value will be the absolute minimum value of function f. This is the key idea to solve the given question. Now, let us start the solution. Let function f is given by fx equal to 2x cube – 24x plus 107. Now, differentiating both sides with respect to x we get fx equal to 6x square – 24. We know derivative of 2x cube is equal to 6x square and derivative of 24x is 24 and derivative of 107 is 0. So, we get f dash x equal to 6x square – 24. Now, for finding critical points we will put f dash x equal to 0. Now, this implies 6x square – 24 is equal to 0. We know f dash x is equal to 6x square – 24. Now, adding 24 on both sides we get 6x square is equal to 24. Now, dividing both sides by 6 we get x square is equal to 4. Now, taking square root on both sides we get x is equal to plus minus 2. Now, we know we have to find the maximum value of function f in the closed interval 1, 3 and here we can see x equal to minus 2 does not belong to the closed interval 1, 3. So, we shall not find the value of function f at x equal to minus 2. Now, we will find the value of function f at x equal to 1 at x equal to 2 and at x equal to 3. Let us first of all find out f1. f1 is equal to 2 multiplied by 1 cube minus 24 multiplied by 1 plus 107. Now, on simplifying we get 2 minus 24 plus 107 which is further equal to 85. So, we get f1 equal to 85. Now, let us find out f2. f2 is equal to 2 multiplied by 2 cube minus 24 multiplied by 2 plus 107. This is equal to 16 minus 48 plus 107. Simplifying we get f2 equal to 75. Now, let us find out f3. f3 is equal to 2 multiplied by 3 cube minus 24 multiplied by 3 plus 107. This is equal to 54 minus 72 plus 107. On simplifying we get f3 equal to 89. Now, this completes the second step of the key idea. Now, we will identify maximum value. Clearly, we can see maximum value of function f is 89 which occurs at x equal to 3. So, we can write maximum value of 2x cube minus 24x plus 107 in rose interval 1, 3 is 89 that occurs at x equal to 3. Now, second part of the question is find the maximum value of the same function in rose interval minus 3 minus 1. Now, we know critical points of the given function are plus 2 and minus 2. Now, we know 2 does not belongs to rose interval minus 3 minus 1. So, we will not find the value of function f at x equal to 2. Now, we will find the value of function f at x equal to minus 3, at x equal to minus 2 and at x equal to minus 1. So, let us find out f minus 3 that is equal to 2 multiplied by minus 3 cube minus 24 multiplied by minus 3 plus 107. Now, this is equal to minus 54 plus 72 plus 107. Now, simply find we get f minus 3 equal to 125. Now, we will find f minus 2 which is equal to 2 multiplied by minus 2 cube minus 24 multiplied by minus 2 plus 107. This is equal to minus 16 plus 48 plus 107. Simplifying we get 139 equal to f minus 2. Now, let us find out f minus 1 which is equal to 2 multiplied by minus 1 cube minus 24 multiplied by minus 1 plus 107. Now, this is equal to minus 2 plus 24 plus 107. Simplifying we get f minus 1 equal to 129. Now clearly we can see maximum value of function f is equal to 139 which occurs at x equal to minus 2. So, we can write maximum value of 2x cube minus 24x plus 107 in closed interval minus 3 minus 1 is 139 that occurs at x equal to minus 2. So, this is our required answer. This completes the session. Hope you understood the solution. Have a nice day.