Lagrange Multipliers

1. Find  the maximum and minimum values of \(f(x,y)=6x+4y\) on the circle \(x^2+y^2=36\).   solution

2. Find  the maximum and minimum values of \(f(x,y)=2xyz\) on the ellipsoid \(x^2+2y^2+3z^2=6\).   solution

3. A rectangular box without a lid is to be made of 12 square meters of cardboard. Find the maximum volume of such a box.    solution 

4. Find the maximum and minimum values of the function \(f(x,y)=x^2+y^2+z^2 \) subject to the constraint \( x^4+y^4+z^4=1 \).   solution 

5. Find the minimum and maximum distances between the ellipse \(x^2+xy+2y^2 =1 \) and the origin.    solution