KKT conditions with two inequality constrains I need to minimize   $f(x,y,z)=x^{2}+2y^{2}+3z^{2}$ subject to
    \begin{align*}
    &x-y-2z\leq 12\\
    &x+2y-3z\leq 8.
   \end{align*} 
So I wrote the lagrangian of $f$.
 \begin{align*}
    \mathcal{L}(x,y,z,\lambda,\mu)=x^{2}+2y^{2}+3z^{2}-\lambda(x-y-2z)-\mu(x+2y-3z)
\end{align*}
I am not sure if the KKT conditions I wrote are right. 
$\frac{\partial \mathcal{L}}{\partial x}=2x-\lambda-\mu=0\\
\frac{\partial \mathcal{L}}{\partial y}=4y+\lambda-2\mu=0\\
\frac{\partial \mathcal{L}}{\partial z}=6z+2\lambda+3\mu=0\\
\lambda(12-x+y+2z)=0\\
\mu(8-x-2y-3z)=0  \\
\lambda,\mu \geq 0\\$
Are the above  kkt conditions correct ? Am I missing any conditions? If these are correct I plan to solve the six equations to find the minimum. Could someone tell me if I am doing this right. Thank you.

It is correct. Just a quick reminder: Note that, in general, for inequality constraint $g(x) \leq 0$, we would consider $g(x)\mu = 0$. Hence, we should have $$\lambda(x-y+2z-12)=0\\
\mu(x+2y+3z-8)=0  .\\$$ But that is the same as what you have written down.
