How would one go about finding the maximum or the minimum value of a function given some constraints; Simple functions are easy to deal with, but what piqued my interest was when I thought about functions having variables multiplied and divided by each other. One such example could be :
What is the maximum value of $x^2y$ give the constraint
$$x + y + \sqrt{2x^2 + 2xy + 3y^2 }=K(constant)$$
and also $x,y\ge0$.
I'm not sure on how to proceed on this one as I've tried a few methods, like taking $x^2y$ to be different variable and use the second derivative method for finding maximum and minimum values. However, no results have been calculated yet.