 Hi and welcome to the session. Let us discuss the following question. Question says, find the local maxima and local minima if any of the following functions. Find also the local maximum and local minimum values as the case may be. Fifth part is fx is equal to x cube minus 6x square plus 9x plus 5. First of all, let us understand if we are given a function f defined on interval i and c belongs to interval i such that f double dash c exists. Then c is a point of local maxima if f dash c is equal to 0 and f double dash c is less than 0 and c is a point of local minima if f dash c is equal to 0 and f double dash c is greater than 0. This is the key idea to solve the given question. In this case, fc is the local maximum value of function f and here fc is the local minimum value of function f. Let us now start with the solution. We are given fx is equal to x cube minus 6x square plus 9x plus 15. Now differentiating both sides with respect to x we get f dash x is equal to 3x square minus 12x plus 9 plus 0. We know derivative of x cube is 3x square derivative of 6x square is 12x derivative of 9x is 9 and derivative of 15 is 0. So we get f dash x is equal to 3x square minus 12x plus 9. Now let us find out all the points at which f dash x is equal to 0. So we will put f dash x is equal to 0. Now this implies 3x square minus 12x plus 9 is equal to 0. Now dividing both sides by 3 we get x square minus 4x plus 3 is equal to 0. Now this implies x square minus 3x minus x plus 3 is equal to 0. We can write minus 4x as minus 3x minus x. Now taking x common from first two terms we get x multiplied by x minus 3 and minus 1 common from last two terms we get minus 1 multiplied by x minus 3 is equal to 0. Now this implies x minus 3 multiplied by x minus 1 is equal to 0. x minus 3 is common in these two terms. So taking x minus 3 common we get x minus 3 multiplied by x minus 1 is equal to 0. Now this implies x minus 3 is equal to 0 or x minus 1 is equal to 0. This implies x is equal to 3 or x is equal to 1. Now to find x is equal to 3 or x is equal to 1 are points of maxima or minima. We will find second derivative of fx we know f dash x is equal to 3x square minus 12x plus 9. Now differentiating both sides with respect to x again we get f double dash x is equal to 6x minus 12. Now we know f dash 3 is equal to 0 and f double dash 3 is equal to 6 multiplied by 3 minus 12 which is equal to 6. So we get f dash 3 is equal to 0 and f double dash 3 is greater than 0. Now this implies 3 is a point of local minima. So we can write x is equal to 3 is a point of local minima. Now we know minimum value at x is equal to 3 is given by f3 f3 is equal to 3q minus 6 multiplied by 3 square plus 9 multiplied by 3 plus 15. This is equal to 27 minus 54 plus 27 plus 15. Simplifying we get f3 is equal to 15. Now we know f dash x is equal to 0 at x is equal to 1 also. So now we will find out f double dash 1 f double dash 1 is equal to 6 multiplied by 1 minus 12 which is equal to minus 6. This is less than 0. So we get f1 is equal to 0 and f double dash 1 is less than 0. This implies x is equal to 1 is a point of local maxima and maximum value is given by f1 which is equal to 1q minus 6 multiplied by 1 square plus 9 multiplied by 1. Plus 15 which is equal to 1 minus 6 plus 9 plus 15. Simplifying we get f1 is equal to 19. So our required answer is local maximum is at x is equal to 1 and local maximum value is equal to 19. Local minima is at x is equal to 3 and local minimum value is equal to 15. This completes the session. Hope you understood the session. Take care and have a nice day.