 Hello and welcome to the session. In this session we will discuss a question which says that if x is real, find the maximum value of 3 plus 2x minus x square. Now let's start with the solution. Here we have to find the maximum value of 3 plus 2x minus x square. Now let us look at 3 plus 2x minus x square is equal to y. This implies 3 plus 2x minus x square minus y is equal to 0, which implies x square minus 2x plus written brackets y minus 3 is equal to 0. Now if x is real then b square minus 4ac is greater than equal to 0. Now from this equation, putting the values of b, a and c here, this implies b here is minus 2, so it will be minus 2 whole square minus 4 into a here is 1 into c here is y minus 3, so it will be y minus 3 greater than equal to 0. This implies 4 minus 4 into y minus 3 the whole is greater than equal to 0. This implies taking 4 common within brackets 1 minus y plus 3 is greater than equal to 0. Further this implies 4 within brackets 4 minus y is greater than equal to 0. Now this condition will be satisfied if 4 and 4 minus y are of same sign. This means here 4 is positive then 4 minus y should be greater than equal to 0. This implies 4 is greater than equal to y which implies y is less than equal to 4. This shows that y cannot be greater than 4 as otherwise 4 into 4 minus y the whole will become negative. This means that if here y will be greater than 4 then this expression will become negative. Hence maximum value of y that is the expression is 4. So this is the solution of the given question and that's all for this session. Hope you all have enjoyed the session.