 Hi and welcome to the session. Let us discuss the following question. Question is, 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. Third part is, Hx is equal to sinx plus cosx where x is greater than 0 and less than pi upon 2. First of all let us understand that if we are given a function f defined on interval i, c belongs to interval i such that f double dash c exists. x is equal to c is a point of local maxima if dash c is equal to 0 and f double dash c is less than 0. Here value of fc is the local maximum value and here fc is the local minimum value. This is the key idea to solve the given question. Now let us start the solution. We are given Hx is equal to 0 and f double dash c is greater than 0. Here value of fc is the local maximum value and here fc is the local minimum value. Now let us start the solution. We are given Hx is equal to sinx plus cosx where x is greater than 0 and less than pi upon 2. Differentiating both the sides with respect to x we get h dash x is equal to cosx minus sinx. We know derivative of sinx is cosx and derivative of cosx is minus sinx. Now we will find all the values of x at which h dash x is equal to 0. So we will put h dash x is equal to 0. Now this implies cosx minus sinx is equal to 0. Now adding sinx on both the sides we get cosx is equal to sinx. Now dividing both the sides by cosx we get 1 is equal to sinx upon cosx. Now we know sinx upon cosx is equal to tanx. So we get 1 is equal to tanx or we can say tanx is equal to 1. Now we know tanx is equal to 1 only when value of x is pi upon 4. So we get x is equal to pi upon 4. Clearly we can see this value of x is greater than 0 and less than pi upon 2. Find the value of h double dash x and x is equal to pi upon 4. First of all we will find out h double dash x. We know h dash x is equal to cosx minus sinx. This we have already shown. Differentiating both the sides with respect to x again we get h double dash x is equal to minus sinx minus cosx. We know derivative of cosx is minus sinx and derivative of sinx is cosx. Now this can be written as minus sinx plus cosx. We can take minus sin common. Now we get h double dash x is equal to minus sinx plus cosx. Now we will find the value of h double dash x and x is equal to pi upon 4. So h double dash pi upon 4 is equal to minus of sin pi upon 4 plus cos pi upon 4. We know sin pi upon 4 is equal to 1 upon root 2 and cos pi upon 4 is also equal to 1 upon root 2. So we get minus of 1 upon root 2 plus 1 upon root 2. This is further equal to minus 2 upon root 2. Now rationalizing we get minus 2 root 2 upon 2, 2 and 2 will cancel each other and we get minus root 2. Now clearly we can see this value is less than 0. Now we get h dash pi upon 4 is equal to 0 and h double dash pi upon 4 is equal to minus root 2 which is less than 0. So this implies there exist maxima at x is equal to pi upon 4 and we know local maximum value is equal to h pi upon 4. h pi upon 4 is equal to sin pi upon 4 plus cos pi upon 4. We know sin pi upon 4 is equal to 1 upon root 2 and cos pi upon 4 is also equal to 1 upon root 2. This can be further written as 2 upon root 2. Now rationalizing we get 2 root 2 upon 2, 2 and 2 will cancel each other and we get h pi upon 4 is equal to root 2. So our required answer is local maximum upper set x is equal to pi upon 4 and local maximum value is equal to root 2. This completes the session. Hope you enjoyed the session. Take care and keep smiling.