In my math lectures, I learnt that an extremum of a function $f:\mathbb R^2\to \mathbb R$ requires $\mathrm{grad}(f)=0$. So if $f$ was $f(x_1, x_2)$ that means $(∂f/∂x_1, ∂f/∂x_2) \cdot (x_1, x_2)^{\top}=0$. (Sorry for my poor Tex skills, am working to improve those).
No I came across the following in an economics lecture and I can't figure out if what they do is correct:
To find an extremum in $\pi=p*f(x_1, x_2) - w_1x_1 - w_2x_2$ they claimed it was sufficient to find a point satisfying $\frac{∂\pi}{∂x_1}=0$ and $\frac{∂\pi}{∂x_2}=0$
In contrast, the normal gradient approach would yield $\frac{∂\pi}{∂x_1}x_1 + \frac{∂\pi}{∂x_2}x_2=0$ as the precondition for an extremum.
It's pretty clear that $\frac{∂\pi}{∂x_1}=0$ and $\frac{∂\pi}{∂x_2}=0$ implies $\frac{∂\pi}{∂x_1}x_1 + \frac{∂\pi}{∂x_2}x_2=0$, but not the other way round. Because of that, I'd say that the approach used in the economics lecture might not find all interesting points for an extremum.
Is that correct? Or is there anything I have overlooked?