Find the maxima and the minima of the given function given the constraint.

$\displaystyle f(x,y)=x^3-y^2$

Constraint:

$\displaystyle x+y=1$

Using the Lagrange Multiplier method, I managed to obtain

$\displaystyle 3x^2=\lambda$

$\displaystyle -2y=\lambda$

Solving for the variables gave me:

$\displaystyle x=\pm \sqrt{\frac{\lambda}{3}}$

$\displaystyle y=-\frac{\lambda}{2}$

So then I plug these values into my constraint to end up with

$\displaystyle \pm \sqrt{\frac{\lambda}{3}}-\frac{\lambda}{2}=1$

Then I end up with the quadratic, after simplifying;

$\displaystyle 3\lambda^2+8\lambda+12=0$

However, this quadratic can't be solved using real numbers. Have I done anything wrong or am I using the wrong approach?

Thank you