Find the maximum value of x²y subject to the constraint:

x+y+(2x²+2xy+3y²)^1/2 =k where k is a constant and x,y>=0

Given,

x+y+(2x² + 2xy + 3y² )^1/2 = k

x,y >= 0

Dear Student,

First write the above relation seperating y on one side and then substituting it in the equation.

z = x²y

1. Find dy/dx

2. Equate it to zero and get the value of x.

3. Then find d²y/dx² & substitute the values of x from Step (2) in d²y/dx²  

^4. If d²y/dx²   is -ve, it is maxima.

5. Substitute that value of y in the function, f = x² y , you get the maximum value.