Consider the function f (x, y) = x3 + y subject to the constraint g (x, 3) = 2x + y - 6 = 0

To analyze the function f(x, y) = x^3 + y subject to the constraint g(x, y) = 2x + y - 6 = 0 using Lagrange multipliers:

1. Calculate the gradients:
\nabla f(x, y) = \left( \frac{\partial f}{\partial x}, \frac{\partial f}{\partial y} \right) = (3x^2, 1)
\nabla g(x, y) = \left( \frac{\partial g}{\partial x}, \frac{\partial g}{\partial y} \right) = (2, 1)

2. Set up the equations for Lagrange multipliers:
\nabla f(x, y) = \lambda \nabla g(x, y)
(3x^2, 1) = \lambda(2, 1)

3. Solve for \lambda and x :
3x^2 = 2\lambda, 1 = \lambda
\lambda = 1
3x^2 = 2
x^2 = \frac{2}{3}
x = \pm \sqrt{\frac{2}{3}}

4. Calculate the corresponding values of y using the constraint equation g(x, y) = 0 :
For x = \sqrt{\frac{2}{3}} , y = 6 - 2\sqrt{\frac{2}{3}}
For x = -\sqrt{\frac{2}{3}} , y = 6 + 2\sqrt{\frac{2}{3}}

Therefore, the critical points are:
(\sqrt{\frac{2}{3}}, 6 - 2\sqrt{\frac{2}{3}}) and (-\sqrt{\frac{2}{3}}, 6 + 2\sqrt{\frac{2}{3}}) .

\boxed{(\sqrt{\frac{2}{3}}, 6 - 2\sqrt{\frac{2}{3}})} and \boxed{(-\sqrt{\frac{2}{3}},6 + 2\sqrt{\frac{2}{3}})}