Use Inequalitites for example AMGM Jensen Cauchy etc Use Ine

Use Inequalitites (for example. AM-GM, Jensen, Cauchy etc.)

Use Inequalitites (for example. AM-GM, Jensen, Cauchy etc.) To maximize the value of if x+y +z = 6 x,y,z > or = to 0 and f(x,y,z) = xy^2 z^3

Solution

AM GM Method

xy² z³     LCM of 1,2 and 3 = 6

x+y+z=6

6x+6y+6z =36

Since we need single power of x , double of y and triple of z , so we break them into single , double and triple respectively

6x + 3y + 3y +2z +2z + 2z =36

(6x .3y . 3y .2z .2z .2z)^1/6 <=36/6

432 xy² z³ <=6⁶

xy² z³ <= 108