As stated in the title, I'm wondering why we need to set the FOC equal to zero in case we have a constrained optimization problem. I know how it works but I don't really get the intuition behind it. I know that might eb a stupid question but I don't like to apply stuff, without knowing why to apply it.
Thanks a lot in advance. :-)
Let’s say that you are trying to find an extremum of $f(\mathbf P)$ along the curve $g(\mathbf P)=\text{constant}$. This will occur at a point $\mathbf P_0$ if $\operatorname df_{\mathbf P_0}[\mathbf v]=0$ for any vector $\mathbf v$ which is tangent to the curve. Since we are constrained to a level curve of $g$, then $\operatorname dg_{\mathbf P_0}[\mathbf v]$ is also zero, so $\operatorname df_{\mathbf P_0}$ must be a scalar multiple of $\operatorname dg_{\mathbf P_0}$, i.e., $\operatorname df_{\mathbf P_0}=\lambda\operatorname dg_{\mathbf P_0}$. Thus, we look for points at which $\operatorname d(f-\lambda g)=0$.
