Consider the function
$$f(x,y) = e^xcos(y).$$
I am asked if this function has a maximum or a minimum inside the unit circle $\{x,y: x^2+y^2<1\}$.
The answer provided here is that, since the partial derivatives of the function are never $0$ simultaneously, the gradient is never zero, so the function has no minima or maxima. The answer is not in English so it would be difficult to quote it exactly here, but the wording implies there is no maxima or minima in any set.
But let's say we included the points $\{x,y:x^2+y^2=1\}$. Wouldn't the function now have a maxima at the "endpoints" $x^2+y^2=1$? To me it seems there is probably a miswording, and the author meant that the function has no maxima on the open set in question, as the function grows arbitrarily large as we approach the edge of the set (while never reaching $x^2+y^2=1$). Or have I misunderstood something?
If you include the cincunference $x^2+y^2=1$, you are looking for minima/maxima of a continuous function over a compact set. Weierstrass's theorem guarantees their existence, and you can compute them using Lagrange multipliers.