Maxima minima questions based on AM GM

Sample questions :

1. X+Y = 60 , what is the maximum value of X x Y ? given that X and Y are positive variables

2. 2x+3y =60 , what is the maximum value of X x Y ? given that X and Y are positive variables

3. X+Y = 60 , what is the maximum value of X^2 xY ? given that X and Y are positive variables

Notes :

1. Rule AM greater than or equal to the GM for a set of numbers

2. When sum is provided for in the question , maximum of the product is asked , and when product is given then Minimum of sum is asked for, this is a result of the nature of the equation. Notice from the A.M point of view it says A.M is greater than or equal to i.e it can not be any less than the GM and hence the minimum , from the GM point of view the same equation would read GM is less than or equal to AM i.e it can not be any more than the AM and thus the maximum .

3. In A>B we get minimum for A and maximum for B

4. The task then in such questions is to balance what is provided in the question and what is asked for in the AM GM framework , essentially the task is that of choosing the right set of variables for which the AM GM rule should be considered , our choices of variables are limited by what is provided by the question and what is asked for .

Practice questions :

1. X+Y+Z = 100 , what is the maximum value of X^2 x Y^3 x Z^5 ? given that X , Y and Z are positive variables

2. 2X +3Y+5Z= 64 , what is the maximum value of X^3 x Y^2 x Z^3 ? given that X , Y and Z are positive variables