minimize $f(x,y,z)=x^2 + y^2+z^2 $ constraint is $x^3+1 \leq 0 $
when I did this using slack variable I get $(x,y,z)=(-1,0,0) $ but it is not working out using lagrange multiplier method.
Please help me out here.
thanks in advance
I understand I will need to use a slack variable to make an equality and will have $x^3+1+S^2=0$
With the Lagrange Multiplier, we check if the constraint is binding or non-binding. So we have:
$L(x, y, z, \lambda) = x^{2} + y^{2} + z^{2} - \lambda(x^{3} + 1)$. We then check our first order conditions:
$2x - 3x^{2} \lambda = 0$
$2y = 0$
$2z = 0$
So if the constraint is binding, then $\lambda \neq 0$, and so $x = -1$. That gives us $\lambda = -\frac{2}{3}$. That gives us $f(-1, 0, 0) = 1$ as the minimum.
And if the constraint is non-binding, then $\lambda = 0$ and $x < -1$, which gives us $x^{2} > 1$.