Find the maximum and minimum values of \(f\left( {x,y} \right) = 10{y^2} - 4{x^2}\) subject to the constraint \({x^4} + {y^4} = 1\).
Find the maximum and minimum values of \(f\left( {x,y} \right) = 3x - 6y\) subject to the constraint \(4{x^2} + 2{y^2} = 25\).
Find the maximum and minimum values of \(f\left( {x,y} \right) = xy\) subject to the constraint \({x^2} - y = 12\). Assume that \(y \le 0\) for this problem. Why is this assumption needed?
Find the maximum and minimum values of \(f\left( {x,y,z} \right) = {x^2} + 3{y^2}\) subject to the constraint \({x^2} + 4{y^2} + {z^2} = 36\).
Find the maximum and minimum values of \(f\left( {x,y,z} \right) = xyz\) subject to the constraint \({x^2} + 2{y^2} + 4{z^2} = 24\).
Find the maximum and minimum values of \(f\left( {x,y,z} \right) = 2x + 4y + {z^2}\) subject to the constraints \({y^2} + {z^2} = 1\) and \({x^2} + {z^2} = 1\).
Find the maximum and minimum values of \(f\left( {x,y,z} \right) = x + y + {z^2}\) subject to the constraints \(x + y + z = 1\) and \({x^2} + {z^2} = 1\).