 Hi and welcome to the session. I am Shashi and I am going to help you with the following question. Question says, find the local maxima and local minima if any of the following functions. Find also the local maximum and the local minimum values as the case may be we are given fx is equal to sin x minus cos x where x is greater than 0 and less than 2 phi. First of all let us understand that if we are given a function f defined on open interval i then let function f be continuous at critical point c in interval i then if f dash x changes sin from positive to negative as x increases to c then c is a point of local maxima that is if f dash x is greater than 0 for x slightly less than c f dash x is less than 0 for x slightly greater than c then c is a point of local maxima and fc is the local maximum value. Now if f dash x is less than 0 for x slightly less than c f dash x is greater than 0 for x slightly greater than c then c is a point of local minima and fc is the local minimum value or we can say if f dash x changes sin from negative to positive as x increases through c then c is a point of local minima. We will use this discussion as our key idea to solve the given question let us now start with the solution we are given fx is equal to sin x minus cos x where x is greater than 0 and less than 2 phi first of all let us find out f dash x differentiating both the sides of this expression with respect to x we get f dash x is equal to cos x minus sin x you know derivative of sin x is cos x and derivative of cos x is minus sin x now simply find we get f dash x is equal to cos x plus sin x now we will find all the points at which f dash x is equal to 0 so we will put f dash x is equal to 0 this implies cos x plus sin x is equal to 0 now dividing both the sides of this expression by cos x we get 1 plus sin x upon cos x is equal to 0 we know sin x upon cos x is equal to tan x so we can write 1 plus tan x is equal to 0 now subtracting 1 from both the sides we get tan x is equal to minus 1 now we know tan 3 pi upon 4 is equal to tan 7 pi upon 4 is equal to minus 1 and clearly we can see values of these two angles lie between 0 and 2 pi so we get x is equal to 3 pi upon 4 or x is equal to 7 pi upon 4 now let us consider the case when x is equal to 3 pi upon 4 when x is slightly less than 3 pi upon 4 f dash x is greater than 0 if x is slightly greater than 3 pi upon 4 f dash x is less than 0 so using key idea we get x is equal to 3 pi upon 4 is a point of local maxima clearly we can see as x increases through 3 pi upon 4 f dash x changes sign from positive to negative so x is equal to 3 pi upon 4 is a point of local maxima and local maximum value is given by f 3 pi upon 4 now f 3 pi upon 4 is equal to sign 3 pi upon 4 minus cos 3 pi upon 4 now we know sign 3 pi upon 4 is equal to 1 upon root 2 and cos 3 pi upon 4 is equal to minus 1 upon root 2 now simply mind we get f 3 pi upon 4 is equal to 2 upon root 2 rationalizing we get f 3 pi upon 4 is equal to root 2 now let us consider the case when x is equal to 7 pi upon 4 now when x is slightly less than 7 pi upon 4 then f dash x is less than 0 when x is slightly greater than 7 pi upon 4 f dash x is greater than 0 now clearly we can see as value of x increases through 7 pi upon 4 f dash x changes its sign from negative to positive so using key idea we get x is equal to 7 pi upon 4 is a point of local minima and local minimum value is given by f 7 pi upon 4 f 7 pi upon 4 is equal to sign 7 pi upon 4 minus cos 7 pi upon 4 sign 7 pi upon 4 is equal to minus 1 upon root 2 and cos 7 pi upon 4 is equal to 1 upon root 2 so substituting their corresponding values in this expression we get f 7 pi upon 4 is equal to minus 2 upon root 2 now rationalizing we get f 7 pi upon 4 is equal to minus root 2 so we get local minimum value is equal to minus root 2 so we get f 7 pi upon 4 is equal to minus root 2 so we get local minimum value is equal to minus root 2 local maximum value is equal to root 2 so our required answer is local maximum occurs at x is equal to 3 pi upon 4 and local maximum value is equal to root 2 local minimum occurs at x is equal to 7 pi upon 4 and local minimum value is equal to minus root 2 so this is our required answer this completes the session hope you understood the solution take care and have a nice day.