Lets consider a function $f$ differentiable (1-times) on $\mathbb{R}\times \mathbb{R}_0^+$. If the restriction $f_{|\mathbb{R}\times \{0\}}$ has in $(0,0)$ a local minimum, what must the function satisfy to have $(0,0)$ as a local minimum without restriction? Means that $f$ has a local minimum in $(0,0)$.
The answer ist $\frac{\delta f}{\delta y}(0,0)>0$.
Does anyone see why?