 Hi and welcome to the session. Let us discuss the following question. Question says, the maximum value of x multiplied by x minus 1 plus 1 whole raise to the power 1 upon 3, where x is greater than equal to 0 and less than equal to 1 is 1 upon 3 raise to the power 1 upon 3, 1 upon 2, 1, 0. Hello, we have to choose the correct answer from A, B, C and D. First of all, let us understand that for finding absolute maximum or absolute minimum values of a function F in a closed interval I, first of all, we will find all critical points of function F in the interval. Now we will find value of function F at all critical points and at end points of the interval we will now identify the maximum and minimum values of function F out of the values calculated in step 2. The maximum value will be the absolute maximum value of function F and minimum value will be the absolute minimum value of function F. So third step to follow is identify maximum and minimum value of function F. This is the key idea to solve the given question. Now let us start with the solution that function F is given by Fx is equal to x multiplied by x minus 1 plus 1 whole raise to the power 1 upon 3 where x is greater than equal to 0 and less than equal to 1. Now differentiating both sides with respect to x we get F dash x is equal to 1 upon 3 multiplied by x multiplied by x minus 1 plus 1 whole raise to the power minus 2 upon 3 multiplied by derivative of x square minus x plus 1. Here we have applied the chain rule to find the derivative of this term. Now this is further equal to 1 upon 3 multiplied by x multiplied by x minus 1 plus 1 whole raise to the power minus 2 upon 3 multiplied by 2x minus 1. Now F dash x can be written as 2x minus 1 upon 3 multiplied by x multiplied by x minus 1 plus 1 whole raise to the power 2 upon 3. Here we have used the rule that x raise to the power minus m is equal to 1 upon x raise to the power m. Here this power was minus 2 upon 3 after taking the reciprocal it is 2 upon 3. Now we will find all the points at which F dash x is equal to 0. That is we will find all the critical points of function f in the given interval. Now F dash x is equal to 0 implies 2x minus 1 upon 3 multiplied by x multiplied by x minus 1 plus 1 whole raise to the power 2 upon 3 is equal to 0. Now multiplying both sides by the term given in the denominator we get 2x minus 1 is equal to 0. Now adding 1 on both sides we get 2x is equal to 1. Now this implies x is equal to 1 upon 2 dividing both sides by 2 we get x is equal to 1 upon 2. Here we have completed first step of the key idea. Now we know x is greater than equal to 0 and less than equal to 1. So now we will find the value of function f at x is equal to 0 at x is equal to 1 and x is equal to 1 upon 2. Let us now find out f 1 upon 2 this is equal to 1 upon 2 multiplied by 1 upon 2 minus 1 plus 1 whole raise to the power 1 upon 3. Now this is equal to 1 upon 2 multiplied by minus 1 upon 2 plus 1 we know 1 upon 2 minus 1 is equal to minus 1 upon 2 whole raise to the power 1 upon 3. Now this is equal to minus 1 upon 4 plus 1 whole raise to the power 1 upon 3 this further implies f 1 upon 2 is equal to minus 1 plus 4 upon 4 whole raise to the power 1 upon 3 or we can say f 1 upon 2 is equal to 3 upon 4 whole raise to the power 1 upon 3. Let us now find out value of function f at x is equal to 0. Now f 0 is equal to 0 multiplied by 0 minus 1 plus 1 whole raise to the power 1 upon 3 this is equal to minus 1 plus 1 whole raise to the power 1 upon 3 simplifying we get f 0 is equal to 0. Now let us find out value of f 1 f 1 is equal to 1 multiplied by 1 minus 1 plus 1 whole raise to the power 1 upon 3 now this is equal to 1 raise to the power 1 upon 3 we know 1 minus 1 is equal to 0 and 0 multiplied by 1 is equal to 0 so we get 0 plus 1 is equal to 1 so we get 1 raise to the power 1 upon 3 we know 1 raise to any power is equal to 1 only so we get f 1 is equal to 1 now clearly we can see out of all the three values of function f maximum value of function f is 1 which occurs at x is equal to 1 so this completes all the three steps of the k idea so our required answer is c so c is our final answer this completes the session hope you understood the session take care and keep smiling