 Hi and welcome to the session. I am Shashi and I am going to help you with the following question. Question says, find both the maximum value and minimum value of 3x raised to the power 4 minus 8x cube plus 12x square minus 48x plus 25 on the closed interval 0,3. First of all, let us understand that if we are given a function f called you know some closed interval AB, then function f has absolute maximum value and absolute minimum value which is attained by function f at least once in the interval AB. This is the key idea to solve the given question. Let us now start the solution. Let us assume that function f is given by fx equal to 3x raised to the power 4 minus 8x cube plus 12x square minus 48x plus 25. Now differentiating both sides with respect to x we get f dash x equal to 12x cube minus 24x square plus 24x minus 48. Now to find the critical value of x we put f dash x equal to 0. Now we get 12x cube minus 24x square plus 24x minus 48 is equal to 0. This implies 12 multiplied by x cube minus 2x square plus 2x minus 4 is equal to 0. Now this implies x cube minus 2x square plus 2x minus 4 is equal to 0. Now in these two terms x square is common so we can write it equal to x square multiplied by x minus 2. In these two terms plus 2 is common so we can write it equal to plus 2 multiplied by x minus 2. Now this implies x minus 2 multiplied by x square plus 2 is equal to 0. Now this further implies x minus 2 is equal to 0 or x square plus 2 is equal to 0. So we get x is equal to 2 or x square is equal to minus 2. Now clearly we can see x square is equal to minus 2. We will give imaginary values of x so we can neglect x square equal to minus 2. Now we get x is equal to 2. Now we will find the values of fx at x equal to 0, 2 and 3. First of all let us find out f0. This is equal to 3 multiplied by 0 raise to the power 4 minus 8 multiplied by 0 cube plus 12 multiplied by 0 square minus 48 multiplied by 0 plus 25. Now we get f0 is equal to 25. Now let us find out f2. f2 is equal to 3 multiplied by 2 raise to the power 4 minus 8 multiplied by 2 cube plus 12 multiplied by 2 square minus 48 multiplied by 2 plus 25. On simplify we get f2 equal to minus 39. Now let us find out f3. f3 is equal to 3 multiplied by 3 raise to the power 4 minus 8 multiplied by 3 cube plus 12 multiplied by 3 square minus 48 multiplied by 3 plus 25. Now on simplify we get 249 minus 216 plus 108 minus 144 plus 25. Now this is further equal to 382 minus 360. We can write it equal to 22. So we get f3 equal to 22. Now clearly we can see absolute minimum value is equal to minus 8 at x equal to 2 and absolute maximum value is equal to 25 at x equal to 0. Now we can write absolute maximum value of f on closed interval 0 3 is 25 at x equal to 0 and absolute minimum value of f on closed interval 0 3 is minus 39 at x equal to 2. Now this is our required answer. Hope you understood the session. Take care and have a nice day.