 geometric mean of a and b of a and b is let's say g okay so a g and b must be in gp that's what we learned last class last session maybe in AP what was happening if c was the you know mean so a c and b were in AP if c was a m right in this case g is gm geometric mean so a gb must mean gp when is that possible when g by a ratio is b by g common ratio so that means g square is a b so geometric mean g is under root so clearly a and b both have to be either positive or negative a and b cannot be one positive and one negative right both of them have to be either positive or negative correct is that understood this is geometric mean of a and b a and b are two positive numbers where you can write where a is greater than zero b is greater than zero and root g is okay or gm okay there's a very famous inequality which is called a and gm inequality so if two numbers are there a and b okay yes so a m is always greater than equal to gm very important I add you know very important inequality always remember this is called a and gm inequality when we'll talk about hm then include hm also in this inequality but always remember a m is a m of any two positive number will always be greater than equal to gm of those two numbers so a plus b by 2 is always greater than equal to root a b so a plus b plus a plus b by 2 is greater than root of a b okay how can you prove this root a plus root b square of any number is greater than equal to 0 yes so let's say root a plus root b square is always greater than 0 is this true statement always it's a square right so square is always greater than equal to 0 am I right so open the brackets you will get what a plus b plus 2 root a b or rather take minus here then it will help you a root a minus root b right is it okay a is this statement okay use the identity a square minus v square a square plus b square minus 2 a b right so that means a plus b is greater than equal to 2 root a b that means a plus b by 2 is greater than equal to root a b always then what happens LHS becomes a plus a by 2 equal to root of a square which is a is equal to a okay so this is very very very very famous and very important inequality so the question is x is greater than 0 y is greater than 0 and z is greater than 0 okay prove that x plus y times y plus z times z plus x is greater than 8 x y z so x plus y by 2 we just learned is greater than 2 sorry x y now equal equality sign is removed because it was mentioned that they are unique distinct x is not equal to y not equal to z is it below are you right so y plus by 2 is greater than root y z and z plus x by 2 is greater than root z x multiply all three what will you get x plus y by 2 into y plus z by 2 into z plus x by 2 right is greater than root of x square y square z square if you multiply you'll get this so what is this x plus y y plus z z plus x is greater than 2 into 2 into 2 8 and x y done no clear right so find the minimum value of expression which one 3 to the power x plus 3 to the power 1 minus x so such kind of questions wherever they find the minimum maximum value you can always use you can try to use inequalities and mgm equality since we have learned it let's apply this so minimum value of this okay now am of 3 to the power x and 3 to the power 1 minus x will be this and this will be greater than equal to what root over 3 to the power x into 3 to the power 1 minus x isn't by mgm inequality so 3 to the power x plus 3 to the power 1 minus x will be greater than equal to 2 times what under root 3 x plus 1 minus x that is this is greater than equal to 2 root 3 so without doing much of this thing we know that this number cannot be lesser than 2 root 3 correct so now how is how does this help and where where all does this help so you know when you plot let's say this was why this is x this is why so hence let this number was 2 root 3 so the graph will always be in this region is it whatever it is it will never come in this region is that clear is that clear right so 3 to the power x plus 3 to the power 1 minus x will always be more than 2 root 3